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TREATISE    ON    ASTRONOMY. 


ELIAS    LOOMIS,    LL.D. 


PLATE  IV 


TREATISE 


ASTEONOMY 


BY 

ELIAS  LOOMIS,  LL.D,, 

PROFESSOR  OF  NATURAL  PHILOSOPHT  AND  ASTRONOMY  IN  TALK  COLLEGE  ;   AUTHOR  OF 

"AN    INTRODUCTION  TO   PRACTICAL  ASTRONOMY,"  AND  OF  A  SERIES 

OF    MATHEMATICS    FOR    SCHOOLS    AND    COLLEGES. 


NEW   YOKK: 

HARPER  &  BROTHERS,  PUBLISHERS, 

327  AND  335  PEARL  STREET. 

1880. 


LOOMIS'S  SERIES  OF  TEXT-BOOKS. 


ELEMENTARY  ARITHMETIC.    166  pp.,  33  cent*. 

TREATISE  Off  ARITHMETIC.    S5J  pp.,  »1  03. 

ELEMENTS  OF  ALGEBRA.    Revised  EdiUon.    J81  pp.,  |1  05. 

AVf  to  Elmntt  of  Algtbra,  for  Use  of  Teachers.    128  pp.,  (1  OS. 
TREATISE  Off  ALGEBRA.    Revised  Edition.    384  pp.,  |1  17. 

Kq  to  Trtatitt  on  Algebra,  for  UM  of  Teacheri.    Sl»  pp.,  $1  IT. 
ALGEBRAIC  PROBLEMS  AND  EXAMPLES.     458  pp.,  $1  05. 
ELEMENTS  OF  GEOMETRY.    Revised  Edition.    388  pp.,  $1  IT. 

ELEMENTS  OF  TRIGONOMETRY,   SURVEYING,  AND  NAVIGATION.    194  rp-,  $1  ". 
TABLES  OF  LOGARITHMS.     150  pp.,  |1  IT. 

Th*  Tngomnttry  and   Takla,  bound  In  one  volume.    860  pp.,  $1  T5. 
ELEMENTS  OF  ANALYTICAL  GEOMETRY.    Revised  Edition.    J61  pp.,|l  IT. 
DIFFERENTIAL  AND  INTEGRAL  CALCULUS.    Revised  Edition.    809  pp.,  $1  IT. 

The  Analytical  Grnnttry  and  Calevliu,  bound  In  one  volume.    570  pp.,  |!  05. 
ELEMENTS  OF  NATURAL  PHILOSOPHY.    351  pp.,  (1  55. 
ELEMENTS  OF  ASTRONOMY.    154  pp.,  $1  IT. 
PRACTICAL  ASTRONOMY.    499  pp.,  *1  T5. 
TREATISE  ON  ASTRONOMY.    S51  pp.,  |1  T5. 
TREATISE  ON  METEOROLOGY.    808  pp.,  $1  T.». 


Entered,  according  to  Act  of  Congress,  in  the  year  18C5,  by 

II  A  KIT.  i:  &  BROTHERS, 
In  the  Clerk's  Office  of  the  Southern  District  of  New  York. 


Engineering  « 

Mathsmatical 

Sciences  3  1  it 

Library 


PREFACE. 


THE  design  of  the  following  treatise  is  to  furnish  a  text-book 
for  the  instruction  of  college  classes  in  the  first  principles  of  As- 
tronomy. My  aim  has  accordingly  been  to  limit  the  book  to 
such  dimensions  that  it  might  be  read  entire  without  omissions, 
and  to  make  such  a  selection  of-topics  as  should  embrace  every 
thing  most  important  to  the  student.  I  have  aimed  to  express 
every  truth  in  concise  and  simple  language ;  and  when  it  was  nec- 
essary to  introduce  mathematical  discussions,  I  have  limited  my- 
self to  the  elementary  principles  of  the  science.  The  entire  book 
is  divided  into  short  articles,  and  each  article  is  preceded  by  a 
caption,  which  is  designed  to  suggest  the  subject  of  the  article. 
Whenever  it  could  be  done  to  advantage,  I  have  introduced  sim- 
ple mathematical  problems,  designed  to  test  the  student's  famil- 
iarity with  the  preceding  principles.  At  the  close  of  the  book 
will  be  found  a  collection  of  miscellaneous  problems,  many  of 
them  extremely  simple,  which  are  to  be  used  according  to  the  dis- 
cretion of  the  teacher. 

I  have  dwelt  more  fully  than  is  customary  in  astronomical 
text-books  upon  various  physical  phenomena,  such  as  the  consti- 
tution of  the  sun,  the  condition  of  the  moon's  surface,  the  phe- 
nomena of  total  eclipses  of  the  sun,  the  laws  of  the  tides,  and  the 
constitution  of  comets.  I  have  also  given  a  few  of  the  results  of 
recent  researches  respecting  binary  stars.  It  is  hoped  that  the 
discussion  of  these  topics  will  enhance  the  interest  of  the  subject 
with  a  class  of  students  who  might  be  repelled  by  a  treatise  ex- 
clusively mathematical. 

My  special  acknowledgments  are  due  to  Professor  H.  A.  New- 
ton, who  has  read  all  the  proofs  of  the  work,  and  to  whom  I  am 
indebted  for  numerous  important  suggestions. 


PREFACE  TO  THE  EDITION  OF  1880. 


WHEN  this  Treatise  was  first  published,  in  1865,  the  horizontal 
parallax  of  the  sun  was  generally  assumed  to  be  8".5776,  accord- 
ing to  the  computations  of  Prof.  Encke  founded  upon  the  tran- 
sits of  Venus  in  1761  and  1769.  Since  that  time  several  new 
determinations  of  this  quantity  have  been  made,  and  they  all  in- 
dicate that  Encke's  value  is  too  small.  The  observations  of  the 
transit  of  Venus  in  1874  are  not  yet  fully  reduced,  and  it  is  not 
certainly  known  what  value  will  best  represent  all  the  observa- 
tions. At  present  it  is  considered  that  the  most  probable  value 
is  S".S5.  When  the  value  of  this  quantity  has  been  definitively 
determined,  all  the  numbers  in  this  Treatise  which  depend  upon 
the  solar  parallax  will  be  changed.  Meanwhile  the  following 
table  is  presented  (deduced  from  a  parallax  of  8".85),  showing 
the  principal  changes  which  may  ultimately  be  expected: 

ELEMENTS  OF  THE  SOLAR  SYSTEM. 


Mean  ])i*mnce  from  the 
tun  In  miles. 

Equatorial  Diame- 
ter in  miles. 

Velocity  in  miles 
per  .»econd. 

Sun  

861  ooo 

Mercury  

35  762  ooo 

2n  56 

Venus 

^yy** 

rj   653 

Earth    .                ... 

18  3g 

Mars  

3  o45 

8  06 

Saturn 

88  1  008  ooo 

09,027 

K  OR 

Uranus  

i  7Ti  685  ooo 

72,759 
35  101 

0.93 

2,774,149.000 

34,5oo 

3.36 

CONTENTS. 


CHAPTER  I. 

T'iB   EARTH — ITS   FIGURE   AND  DIMENSIONS. — DENSITY. — ROTATION. 

P*ft 

The  Phenomena  of  the  Diurnal  Motion 9 

The  Figure  of  the  Earth— how  determined 15 

Dimensions  of  the  Earth — how  determined 16 

The  Celestial  and  Terrestrial  Spheres 18 

Effects  of  Centrifugal  Force  upon  the  Form  of  the  Earth 22 

Measurement  of  an  Arc  of  the  Meridian 25 

The  Density  of  the  Earth — how  determined 28 

Direct  Proof  of  the  Earth's  Rotation 32 

Artificial  Globes— Problems  on  the  Terrestrial  Globe 35 

CHAPTER  II. 

THE   PRINCIPAL   ASTRONOMICAL   INSTRUMENTS. 

The  Astronomical  Clock— its  Error  and  Rate 38 

The  Transit  Instrument— its  Adjustments 39 

The  Mural  Circle— Reading  Microscope 43 

The  Altitude  and  Azimuth  Instrument 47 

The  Sextant— its  Adjustments  and  Use 49 

CHAPTER  III. 

ATMOSPHERIC   REFRACTION.— TWILIGHT. 

The  Law  of  Atmospheric  Refraction 52 

Refraction  determined  by  Observation 53 

The  Cause  of  Twilight— its  Duration » 56 

CHAPTER  IV. 

EARTH'S  ANNUAL  MOTION. — EQUATION  OF  TIME. — CALENDAR. 

The  Sun's  apparent  Motion — the  Equinoxes,  Solstices,  etc 58 

The  Change  of  Seasons — its  Cause 62 

The  Form  of  the  Earth's  Orbit 64 

Sidereal  and  Solar  Time— Mean  Time  and  Apparent  Time 67 

The  Equation  of  Time  explained 69 

The  Calendar— Julian  and  Gregorian 72 

Problems  on  the  Celestial  Globe .' 74 


^  i  CONTEXTS. 


CHAPTER  V. 

PARALLAX.— ASTRONOMICAL  PROBLEMS.  ?^ 

Diurnal  Parallax— Horizontal  Parallax 76 

How  to  determine  the  Parallax  of  the  Moon 77 

Astronomical  Problems— Latitude— Time,  etc * 

To  liuJ  the  Time  of  the  Sun's  Rising  or  Setting 83 

CHAPTER  VI. 

THE    BUN — ITS   PHYSICAL   CONSTITUTION. 

Distance  and  Diameter  of  the  Sun 


The  Physical  Constitution  of  the  Sun 90 

Theory  of  the  Constitution  of  the  Sun 94 

The  Zodiacal  Light  described 99 

CHAPTER  VII. 

PRECESSION  OF   THE    EQUINOXES. — NUTATION. — ABERRATION. 

Precession  of  the  Equinoxes — its  Cause 100 

Nutation,  Solar  and  Lunar 104 

Aberration  of  Light — its  Cause 105 

Line  of  the  Apsides  of  the  Earth's  orbit 107 

CHAPTER  VIII. 

THE    MOON — ITS  MOTION — PHASES — TELESCOPIC    APPEARANCE. 

Distance,  Diameter,  etc.,  of  the  Moon 109 

Phases  of  the  Moon — Harvest  Moon 113 

Has  the  Moon  an  Atmosphere  ? 116 

Telescopic  appearance  of  the  Moon 118 

Libretions  of  the  Moou— Changes  of  the  Moon's  Orbit 124 

CHAPTER  IX. 

CENTRAL   FORCES. — GRAVITATION. — LUNAR    IRREGULARITIES. 

Curvilinear  Motion — Kepler's  Laws 128 

Theorems  respecting  Motion  in  an  Orbit 129 

Motion  in  an  Elliptic  Orbit 132 

The  Law  of  Gravitation— Motions  of  Projectiles 137 

The  Problem  of  the  Three  Bodies 142 

Sun's  disturbing  Force  computed 146 

Evection,  Variation,  Annual  Equation,  etc 148 

Motion  of  the  Moon's  Nodes 150 

CHAPTER  X. 

ECLIPSES   OF   THE    MOON. 

Dimensions,  etc.,  of  the  Earth's  Shadow 153 

Lunar  Ecliptic  Limits — how  determined 15o 

The  Earth's  Penumbra— its  Dimensions 157 

The  Computation  of  Lunar  Eclipses 159 

Computation  illustrated  by  an  Example 161 


CONTENTS.  vil 

CHAPTER  XI. 

ECLIPSES   OP   THE    SUN. 

Pmgt 

Dimensions,  etc.,  of  the  Moon's  Shadow , 166 

Different  Kinds  of  Eclipses  of  the  Sun 169 

Phenomena  attending  Eclipses  of  the  Sun : 173 

Corona — Baily's  Beads — Flame-like  Protuberances 1 75 

CHAPTER  XII. 

METHODS   OF   FINDING   THE    LONGITUDE    OF   A  PLACE. 

Method  of  Chronometers  explained 179 

Method  by  Eclipses,  Occultations,  Lunar  Distances,  etc 181 

Method  by  the  Electric  Telegraph— Velocity  of  Electric  Fluid 182 

CHAPTER  XIII. 

THE   TIDES. 

Definitions — Cause  of  the  Tides 185 

Cotidal  Lines— Velocity  of  Tidal  Wave 188 

The  Tides  modified  by  Conformation  of  the  Coast , 192 

The  Diurnal  Inequality  in  the  Height  of  the  Tides 194 

Tides  of  the  Pacific  Ocean,  Gulf  of  Mexico,  etc 196 

CHAPTER  XIV. 

THE   PLANETS — ELEMENTS    OF   THEIR   ORBITS. 

Number  of  the  Planets— The  Satellites 198 

Apparent  Motions  of  the  Planets  explained 200 

The  Elements  of  the  Orbit  of  a  Planet 204 

To  determine  the  Distance  of  a  Planet  from  the  Sun 207 

To  determine  the  Position  of  the  Nodes— Inclination  of  Orbit,  etc 209 

CHAPTER  XV. 

THE    INFERIOR   PLANETS. 

Greatest  Elongations — Phases 213 

Mercury  and  Venus — their  Periods,  Distances,  etc 214 

Transits  of  Mercury  and  Venus  across  the  Sun's  Disc 217 

Sun's  Parallax — how  determined 219 

CHAPTER  XVI. 

THE    SUPERIOR  PLANETS. 

Mars — Distance — Phases — Form,  etc 222 

The  Minor  Planets — Discovery — Number,  etc 224 

Jupiter — Distance — Diameter — Belts,  etc , 227 

Jupiter's  Satellites— Distances— Eclipses— Occultations,  etc 228 

Velocity  of  Light— how  and  by  whom  discovered 232 

Saturn — Distance — Diameter — Rotation 233 

His  Rings — their  Disappearance  explained 234 

Saturn's  Satellites— their  Number,  Distance,  etc ^....  239 

Uranus — Discovery — Distance — Diameter,  etc , 240 

Neptune— History  of  its  Discovery— its  Satellite 343 


yjjj  CONTENTS. 

CHAPTER  XVII. 

QUANTITY   OF   MATTER  IN  THE   SCN   AND  PLANETS.-PLANETARY   PERTURBATIONS^ 

How  to  determine  the  Mass  of  a  Planet 2*7 

The  Perturbations  of  the  Planets— how  computed 2a 

The  Secular  Inequalities  of  the  Planets 253 

The  Stability  of  the  Solar  System *>5 

CHAPTER  XVIII. 

COMETS.— COMETARY   ORBITS.— SHOOTING    STARS. 

Number  of  Comets— the  Coma— Nucleus— Tail,  etc 257 

Comets' Tails— Dimensions— Position 260 

Cometary  Orbits— how  computed 265 

Halley's  Comet— its  History  and  Peculiarities 269 

Encke's  Comet— Hypothesis  of  a  resisting  Medium 271 

Biela's  Comet— Faye's  Comet— Brorsen's  Comet,  etc 272 

Comet  of  1744— Comet  of  1770— Comet  of  1843,  etc 276 

Shooting  Stars — Detonating  Meteors — jErolites 279 

CHAPTER  XIX. 

THE  FIXED   STARS — THEIR   LIGHT,  DISTANCE,  AND   MOTIONS. 

Classification  of  the  Fixed  Stars— their  Brightness 284 

The  Constellations — How  Stars  are  Designated 287 

Periodic  Stars — Cause  of  this  Periodicity — Temporary  Stars 290 

Distance  of  the  Fixed  Stars — Parallaxes  determined 292 

Proper  Motion  of  the  Stars— Motion  of  the  Solar  System 295 

CHAPTER  XX. 

DOUBLE   STARS. — CLUSTERS. — NEBULffi. 

Double  Stars— Colored  Stars— Stars  optically  Double 297 

Binary  Stars — Gamma  Virginis — Alpha  Centauri,  etc 299 

Mass  of  a  Binary  Star  computed— Triple  Stars 303 

Clusters  of  Stars— Nebulae 304 

Planetary  Nebula— Variable  Nebula? 308 

The  Milky  Way— its  Constitution  and  Extent 310 

The  Nebular  Hypothesis— how  tested 314 

MISCELLANEOUS  PROBLEMS 316 

TABLES — ELEMENTS  OF  THE  PLANETS,  etc 321 

EXPLANATION  OF  THE  TABLES 333 

EXPLANATION  OF  THE  PLATES 337 


ASTRONOMY. 


CHAPTER  I. 

GENERAL  PHENOMENA  OF  THE  HEAVENS. — FIGURE  AND  DISTEN- 
SIONS OF  THE  EARTH. — DENSITY  OF  THE  EARTH. — PROOF  OF 
THE  EARTH'S  ROTATION. — ARTIFICIAL  GLOBES. 

•f  1.  Astronomy  is  the  science  which  treats  of  the  heavenly  bodies. 
The  heavenly  bodies  consist  of  the  sun,  the  planets  with  their  satel- 
lites, the  comets,  and  tfie  fixed  stars. 

Astronomy  is  divided  into  Spherical  and  Physical.  Spherical 
Astronomy  treats  of  the  appearances,  magnitudes,  motions,  and 
distances  of  the  heavenly  bodies.  Physical  Astronomy  applies 
the  principles  of  Mechanics  to  explain  the  motions  of  the  heav- 
enly bodies,  and  the  laws  by  which  they  are  governed. 

2.  Diurnal  motion. — If  we  examine  the  heavens  on  a  clear 
night,  we  shall  soon  perceive  that  the  stars  constantly  maintain 
the  same  position  relative  to  each  other.  A  map  showing  the 
relative  position  of  these  bodies  on  any  night,  will  represent 
them  with  equal  exactness  on  any  other  night.  They  all  seem 
to  be  at  the  same  distance  from  us,  and  to  be  attached  to  the 
surface  of  a  vast  hemisphere,  of  which  the  place  of  the  observer 
is  the  centre.  But,  although  the  stars  are  relatively  fixed,  the 
hemisphere,  as  a  whole,  is  in  constant  motion.  Stars  rise  oblique- 
ly from  the  horizon  in  the  east,  cross  the  meridian,  and  descend 
obliquely  to  the  west.  The  whole  celestial  vault  appears  to  be 
in  motion  round  a  certain  axis,  carrying  with  it  all  the  objects 
visible  upon  it,  without  disturbing  their  relative  positions.  •  The 
point  of  the  heavens  which  lies  at  the  extremity  of  this  axis  of 
rotation  is  fixed,  and  is  called  the  pole.  There  is  a  star  called 
the  pole  star,  distant  about  1£°  from  the  pole,  which  moves  in  a 
small  circle  round  the  pole  as  a  centre.  All  other  stars  appear 


!()  ASTRONOMY. 

also  to  be  carried  around  the  pole  in  circles,  preserving  always 
the  same  distance  from  it. 

3.  Axis  of  the  celestial  sphere.     In  order  to  determine  the  posi- 
tion of  the  axis  of  rotation,  take  a  telescope  whose  field  of  view 
is  at  least  three  degrees  in  diameter,  and  direct  it  in  such  a  man- 
ner that  the  pole  star  may  be  seen  throughout  its  diurnal  motion 
without  changing  the  position  of  the  telescope.     The  stay  will 
describe  a  small  circle  around  the  centre  of  the  field  of  view  of 
the  telescope,  and  this  centre  is  the  pole  of  the  heavens.     The 
surface  of  the  sphere  to  which  all  the  celestial  bodies  appear  to 
be  attached  is  called  the  celestial  sphere,  and  the  axis  of  rotation 
is  the  axis  of  the  celestial  sphere. 

4.  Use  of  a  telescope  mounted  equatorially.— Raving  determined 
the  axis  of  the  celestial  sphere,  a  telescope  may  be  mounted  ca- 
pable of  revolving  upon  a  fixed  axis  which  points  toward  the 
celestial  pole,  in  such  a  manner  that  the  telescope  may  be  placed 
at  any  desired  angle  with  the  axis,  and  there  may  be  attached  to 
it  a  graduated  circle  by  which  the  magnitude  of  this  angle  may 
be  measured.     A  telescope  thus  mounted  is  called  an  equatorial 
telescope,  and  it  is  frequently  connected  with  clock-work,  which 
gives  it  a  motion  round  the  axis  corresponding  with  the  rotation 
of  the  celestial  sphere. 

5.  Diurnal  paths  of  the  heavenly  bodies. — Let  now  the  telescope 
be  directed  to  any  star  so  that  it  shall  be  seen  in  the  centre  of 
the  field  of  view,  and  let  the  clock-work  be  connected  with  it  so 
as  to  give  it  a  perfectly  uniform  motion  of  rotation  from  east  to 
west.     The  star  will  follow  the  telescope,  and  the  velocity  of  mo- 
tion may  be  so  regulated,  that  the  star  shall  remain  in  the  centre 
of  the  field  of  view  from  rising  to  setting,  the  telescope  all  the 
time  maintaining  the  same  angle  with  the  axis  of  the  heavens. 
The  same  will  be  true  of  every  star  to  which  the  telescope  is  di- 
rected; from  which  we  conclude  that  all  objects  upon  the  firma- 
ment describe  circles  at  right  angles  to  its  axis,  each  object  al- 
ways remaining  at  the  same  distance  from  the  pole. 

6.  Time  of  one  revolution  oftJie  celestial  sphere.- — If  the  telescope 
be  detached  from  the  clock-work,  and,  having  been  pointed  upon 


GENERAL  PHENOMENA  OF  THE  HEAVENS.        11 

a  star,  be  left  fixed  in  its  position,  and  the  exact  time  of  the  star's 
passing  the  central  wire  be  noted,  on  the  next  night  at  about  the 
same  hour  the  star  will  again  arrive  upon  the  central  wire.  The 
time  elapsed  between  these  two  observations  will  be  found  to  be 
23h.  56m.  4s.,  expressed  in  solar  time. 

This,  then,  is  the  time  in  which  the  celestial  sphere  makes  one 
revolution ;  and  this  time  is  always  the  same,  whatever  be  the 
star  to  which  the  telescope  is  directed. 

7.  A  sidereal  day. — The  time  of  one  complete  revolution  of  the 
firmament  is  called  a  sidereal  day.     This  interval  is  divided  into 
24  sidereal  hours,  each  hour  into  60  minutes,  and  each  minute 
into  60  seconds. 

Since  the  celestial  sphere  turns  through  360°  in  24  sidereal 
hours,  it  turns  through  15°  in  one  sidereal  hour,  and  through  1° 
in  four  sidereal  minutes. 

8.  The  diurnal  motion  is  never  suspended. — With  a  telescope  ot 
considerable  power,  all  the  brighter  stars  can  be  seen  throughout 
the  day,  unless  very  near  the  sun ;  and  by  the  method  of  obser« 
vation  already  described,  we  find  that  the  same  rotation  is  pre* 
served  during  the  day  as  during  the  night. 

All  the  heavenly  bodies,  without  exception,  partake  of  this  di- 
urnal motion ;  but  the  sun,  the  moon,  the  planets,  and  the  comets 
appear  to  have  a  motion  of  their  own,  by  which  they  change 
their  position  among  the  stars  from  day  to  day. 

9.  The  celestial  equator  is  the  great  circle  in  which  a  plane  pass- 
ing through  the  earth's  centre,  and  perpendicular  to  the  axis  of 
the  heavens,  intersects  the  celestial  sphere. 

10.  If  a  plummet  be  freely  suspended  by  a  flexible  line  and 
allowed  to  come  to  a  state  of  rest,  this  line  is  called  a  vertical  line. 
The  point  where  this  line  produced  meets  the  visible  half  of  the 
celestial  sphere,  is  called  the  zenith  ;  and  the  point  where  it  meets 
the  invisible  hemisphere,  which  is  under  the  plane  of  the  horizon, 
is  called  the  nadir. 

Every  plane  passing  through  a  vertical  line  is  called  a  vertical 
plane,  or  a  vertical  circle. 

That  vertical  circle  which  passes  through  the  celestial  pole  is 


12  ASTRONOMY. 

called  Hie  meridian.    The  vertical  circle  at  right  angles  to  the 
meridian  is  called  the  prime  vertical. 

11.  A  horizontal  plane  is  a  plane  perpendicular  to  a  vertical 
line. 

The  sensible  horizon  of  a  place  is  the  circle  in  which  a  plane 
passing  through  the  place,  and  perpendicular  to  the  vertical  line 
at  the  place,  cuts  the  celestial  sphere. 

The  rational  horizon  is  the  circle  in  which  a  plane  passing 
through  the  earth's  centre,  and  parallel  to  the  sensible  horizon, 
cuts  the  celestial  sphere.  On  account  of  the  distance  of  the  stars, 
these  two  planes  intersect  the  celestial  sphere  sensibly  in  the 
same  great  circle. 

The  meridian  and  prime  vertical  meet  the  horizon  in  four 
points,  called  the  cardinal  points;  or  the  nortfi,  south,  east,  and 
west  points. 

12.  The  altitude  of  a  heavenly  body  is  its  elevation  above  the 
horizon  measured  on  a  vertical  circle.     The  zenith  distance  of  a 
body  is  its  distance  from  the  zenith  measured  on  a  vertical  circle. 
The  zenith  distance  is  the  complement  of  the  altitude. 

The  azimuth  of  a  body  is  the  arc  of  the  horizon  intercepted 
between  the  north  or  south  point  of  the  horizon,  and  a  vertical 
circle  passing  through  the  body.  Altitudes  and  azimuths  are 
measured  in  degrees,  minutes,  and  seconds.  The  amplitude  of  a 
star  is  its  distance  from  the  east  or  west  point  at  the  time  of  its 
rising  or  setting.  • 

13.  Consequences  of  the  diurnal  motion. — If  an  observer  could 

watch  the  whole  apparent 
path  of  any  star  in  the  sky, 
he  would  see  it  describe  a 
circle  around  the  line  PP'; 
but  as  only  half  the  celestial 
sphere  is  visible,  it  is  evident 
JN  that  a  part  of  the  path  of  a 
star  may  lie  below  the  hori- 
zon and  be  invisible.  Thus, 
in  Fig.  1,  let  PP'  be  the  axis 
of  rotation  of  the  celestial 
sphere;  NILSMK  be  the  ho. 
mon  produced  to  intersect 


GENERAL  PHENOMENA  OF  THE  HEAVENS.        13 

the  sphere,  and  dividing  it  into  two  hemispheres,  NS  being  the 
north  and  south  line.  If  the  parallel  circles  passing  through 
A,  C,  E,  and  G  be  the  apparent  diurnal  paths  of  four  stars,  then 
it  is  evident  that 

1st.  The  star  which  describes  the  circle  AB  will  never  descend 
below  the  horizon. 

2d.  The  star  which  describes  the  circle  GH  will  never  come 
above  the  horizon. 

3d.  The  star  which  describes  the  circle  ICKD  will  be  above 
the  horizon  while  it  moves  through  ICK,  and  below  the  horizon 
through  the  portion  KDL 

4th.  The  star  which  describes  the  circle  LEMF  will  be  above 
the  horizon  through  the  portion  of  the  circle  LEM,  and  below 
the  horizon  through  the  portion  MFL. 

These  stars  are  said  to  rise  at  I  and  L,  and  to  set  at  K  and  M. 
They  rise  in  the  eastern  part  of  the  horizon,  and  set  in  the 
western. 

"With  the  star  C,  the  visible  portion  of  its  path  ICK  is  greater 
than  the  invisible  portion  KDI ;  while  with  the  star  E,  the  visible 
portion  of  its  path  LEM  is  less  than  the  invisible  portion  MFL. 

14.  Culminations  of  the  heavenly  bodies. — "When  stars  cross  the 
meridian  above  the  pole  they  are  said  to  culminate,  or  attain 
their  greatest  altitude.     All  stars  cross  the  meridian  twice  every 
day;  once  above  the  pole,  and  once  below  the  pole.     The  for- 
mer is  called  their  upper  culmination,  the  latter  is  called  their 
lower  culmination.     Thus  the  star  which  describes  the  circle  AB 
has  its  upper  culmination  at  A,  and  its  lower  culmination  at  B. 

It  is  evident  from  the  figure  that  all  stars  which  lie  to  the 
north  of  the  equator,  will  remain  above  the  horizon  for  a  longer 
period  than  below  it ;  all  stars  south  of  the  equator  will  remain 
above  the  horizon  for  a  shorter  time  than  below  it;  and  stars 
situated  in  the  plane  of  the  equator  will  remain  above  the  hori- 
zon and  below  it  for  equal  periods  of  time. 

15.  How  the  pole  star  may  be  found. — Among  the  most  remark- 
able of  the  stars  which  never  set  in  the  latitude  of  New  York,  is 
the  group  of  stars  known  as  Ursa  Major,  shown  in  Fig.  2,  which 
also  represents   the   constellations  Ursa  Minor  and  Cassiopea. 
The  constellation  Ursa  Major  (represented  on  the  left),  is  easily 


J4  ASTRONOMY. 

recognized  by  its  resemblance  to  the  figure  of  a  dipper,  and  may 
be  used  to  find  the  pole  star  by  drawing  a  line  through  /3  and  a 

Fig.  2. 


(called  the  Pointers),  which  will  pass  through  the  pole  star  a 
Ursaj  Minoris.  A  line  drawn  through  8  Ursse  Majoris  and  the 
pole  star,  will  pass  nearly  through  /3  Cassiopeas  (represented  on 
the  right). 

16.  What  stars  never  set. — If  a  circle  were  drawn  through  N, 
the  north  point  of  the  horizon,  parallel  to  the  equator,  it  would 
cut  off  a  portion  of  the  celestial  sphere  having  P  for  its  centre, 
all  of  which  would  be  above  the  horizon ;  and  a  circle  drawn 
through  S,  the  south  point  of  the  horizon,  parallel  to  the  equator, 
would  cut  off  a  portion  having  P'  for  its  centre,  which  would  be 
wholly  below  the  horizon.     Stars  which  are  nearer  to  the  visible 
pole  than  the  point  N  never  set,  and  those  which  are  nearer  to 
the  invisible  pole  than  the  point  S  never  rise. 

17.  Why  a  "knowledge  of  the  dimensions  of  the  earth  is  important. — 
The  bodies  of  which  astronomy  treats  are  all  (with  the  exception 
of  the  earth)  inaccessible.     Hence,  for  determining  their  distances, 
we  are  obliged  to  employ  indirect  methods.     The  eye  can  only 
judge  of  the  direction  of  objects,  and  is  unable  to  determine  di- 
rectly their  distances ;  but  by  measuring  the  bearings  of  an  inac- 
cessible object  from  two  points  whose  distance  from  each  other  is 
known,  we  may  compute  the  distance  of  that  object  by  the  meth- 
ods of  trigonometry.    In  all  our  observations  for  determining  the 
distance  of  the  celestial  bodies,  the  base  line  must  be  drawn  upon 
the  earth.     It  is  therefore  necessary  to  determine  with  the  ut- 
most precision  the  form  and  dimensions  of  the  earth. 


FIGURE   OF  THE   EARTH.  15 

18.  Proof  that  the  earth  is  globular.— The  figure  of  the  earth  is 
nearly  globular.  This  is  proved, 

1st.  By  its  having  been  many  times  sailed  round  in  different 
directions.  This  fact  can  only  be  explained  by  supposing  that 
the  earth  is  rounded ;  but  it  does  not  alone  furnish  sufficiently 
precise  information  of  its  exact  figure. 

2d.  By  the  phenomena  of  eclipses  of  the  moon.  These  eclipses 
are  caused  by  the  earth  coming  between  the  sun  and  moon,  so 
as  to  cast  its  shadow  upon  the  latter.  The  form  of  this  shadow 
is  always  such  as  one  globe  would  project  upon  another.  Hence 
we  must  admit  that  the  earth  is  of  a  globular  form  unless  we 
deny  that  the  eclipse  is  caused  by  the  earth's  shadow. 

3d.  By  our  seeing  the  top-mast  of  a  ship,  as  it  recedes  from  the 
observer,  after  the  hull  has  disappeared.  If  the  earth  was  a  plane 
surface,  the  top-mast,  having  the  smallest  dimensions,  should  dis- 
appear first,  while  the  hull  and  sails,  having  the  greatest  dimen- 
sions, should  disappear  last ;  but,  in  fact,  the  reverse  takes  place. 


Land  is  visible  from  the  top-mast  when  it  can  not  be  seen  from 
the  deck.  The  tops  of  mountains  can  be  seen  from  a  distance 
when  their  base  is  invisible.  The  sun  illumines  the  summits  of 
mountains  long  after  it  has  set  in  the  valleys.  An  asronaut,  as- 
cending in  his  balloon  after  sunset,  has  seen  the  sun  reappear 
with  all  the  effects,  of  sunrise ;  and  on  descending,  he  has  wit- 
nessed a  second  sunset. 

4th.  If  we  travel  northward,  following  a  meridian,  we  shall  find 
the  altitude  of  the  pole  to  increase  continually  at  the  rate  of  one 
degree  for  a  distance  of  about  69  miles.  This  proves  that  a  sec- 
tion of  the  earth  made  by  a  meridian  plane  is  very  nearly  a  circle, 
and  also  affords  us  the  means  of  determining  its  dimensions,  as 
shown  in  Art.  20. 


1(J  ASTRONOMY. 

19.  First  method  of  determining  the  earths  diameter.— The  facts 
just  stated  not  only  demonstrate  that  the  earth  is  globular,  but 
afford  us  a  rude  method  of  computing  its  diameter.     For  this 
purpose  we  measure  the  height  of  some  mountain,  and  also  the  dis- 
tance at  which  it  can  be  seen  at  sea.    Let  BD  represent  a  mountain 

FiR  4  (Chimborazo,  for  example),  4  miles  in  height ; 

and  suppose  the  distance,  AB,  at  which  it  can 
be  seen  at  sea,  is  179  miles.  Then,  in  the  tri- 
angle ABC,  representing  the  radius  of  the 
earth  by  R,  we  shall  have 

(R+4)2=R2+1792, 

from  which  we  find  that  R— 4000  miles  nearly. 
Thus  we  learn  that  the  radius  of  the  earth  is 
about  4000  miles.  Similar  observations  made 
in  all  parts  of  the  earth,  give  nearly  the  same  value  for  the  ra- 
dius, which  can  only  be  explained  by  supposing  that^the  earth 
is  nearly  a  sphere. 

The  earth  is  known  to  be  globular  by  the  most  accurate  meas- 
urements, as  will  be  more  fully  explained  hereafter. 

20.  Second  method  of  determining  the  earths  diameter. — Having 
ascertained  the  general  form  of  the  earth,  we  wish  to  determine, 
as  accurately  as  we  can,  its  diameter.     For  this  purpose  we  first 
ascertain  the  length  of  one  degree  upon  its  surface ;  that  is,  the 
distance  between  two  points  on  the  earth's  surface  so  situated 

Fig  5      that  the  lines  drawn  from  them  to  the  centre  of  the  earth 
^S'  may  make  with  each  other  an  angle  of  one  degree. 

Let  P  and  P'  be  two  places  on  the  earth's  surface,  dis- 
tant from  each  other  about  70  miles,  and  let  C  be  the 
centre  of  the  earth.  Suppose  two  persons  at  the  places 
P  and  P'  observe  two  stars  S  and  S',  which  are  at  the 
same  instant  vertically  over  the  two  places — that  is,  in 
the  direction  of  plumb-lines  suspended  at  those  places. 
Let  the  directions  of  these  plumb-lines  be  continued 
downward  so  as  to  intersect  at  C  the  centre  of  the  earth. 
The  angle  which  the  directions  of  these  stars  make  at 
P  is  SPS',  and  the  angle  as  seen  from  C  is  SCS' ;  but, 
on  account  of  the  distance  of  the  stars,  these  angles  are 
sensibly  equal  to  each  other.  If,  then,  the  angle  SPS' 
be  measured,  and  the  distance  between  the  places  P  and 


FIGURE   AND   DIMENSIONS   OF  THE   EARTH.  17 

P'  be  also  measured  by  the  ordinary  methods  of  surveying,  the 
length  of  one  degree  can  be  computed.  In  this  way  it  has  been 
ascertained  that  the  length  of  a  degree  of  the  earth's  surface  is 
about  69  statute  miles,  or  365,000  feet. 

Since  a  second  is  the  3600th  part  of  a  degree,  it  follows  that 
the  length  of  one  second  is  one  hundred  feet  very  nearly. 

Since  the  plumb-line  is  perpendicular  to  the  earth's  surface,  its 
change  of  direction  in  passing  from  one  place  to  another  may  be 
found  by  allowing  one  second  for  every  hundred  feet,  or  more 
exactly  by  allowing  865,000  feet  for  each  degree. 

21.  The  circumference  of  the  earth  may  be  found  approximately 
by  the  proportion 

1  degree  :  360  degrees  : :  69  miles  :  24,840  miles ; 
and  hence  the  diameter  is  found  to  be  about  7900  miles ;  which 
results  are  a  little  too  small,  but  may  be  employed  as  convenient 
numbers  for  illustration. 

The  earth  being  globular,  it  is  evident  that  the  terms  up  and 
down  can  not  every  where  denote  the  same  absolute  direction. 
The- term  up  simply  denotes  from  the  earth's  centre,  while  down 
denotes  towards  the  earth's  centre ;  but  the  absolute  direction  de- 
noted by  these  terms  at  New  York  is  very  different  from  that 
denoted  by  the  same  terms  at  London  or  Canton. 

22.  Irregularities  of  the  earth's  surface. — The  highest  mountain 
peaks  do  not  exceed  five  miles  in  height,  which  is  about  -n^nr  °f 
the  earth's  diameter.     Accordingly,  on  a  globe  16  inches  in  di- 
ameter, the  highest  mountain  peak  would  be  represented  by  a 
protuberance  having  an  elevation  of  T~y  inch,  which  is  about 
twice  the  thickness  of  an  ordinary  sheet  of  writing-paper.     The 
general  elevation  of  the  continents  above  the  sea  would  be  cor- 
rectly represented  by  the  thinnest  film  of  varnish.     In  other 
words,  the  irregularities  of  the  earth's  surface  are  quite  insignifi- 
cant in  comparison  with  its  absolute  dimensions. 

23.  Cause  of  the  diurnal  motion. — The  apparent  diurnal  rotation 
of  the  heavens  may  be  caused  either  by  a  real  motion  of  the  ce- 
lestial sphere,  or  by  a  real  motion  of  the  earth  in  a  contrary  di- 
rection.    The  former  supposition  is  felt  to  be  absurd  as  soon  as 
we  learn  the  distances  and  magnitudes  of  the  celestial  bodies. 

B 


18  ASTRONOMY. 

The  latter  supposition  is  in  itself  not  improbable,  and  perfectly 
explains  all  the  phenomena.  Moreover,  we  find  direct  proof  of 
the  rotation  of  the  earth,  in  the  descent  of  a  body  falling  from 
a  great  height,  which  falls  a  little  to  the  eastward  of  a  vertical 

The  figure  of  the  earth,  which  is  not  that  of  a  perfect  sphere,  af- 
fords independent  proof  of  its  rotation. 

Analogy  also  favors  the  same  conclusion.  All  the  planets 
which  we  have  been  able  satisfactorily  to  observe,  rotate  on  their 
axes,  and  their  figures  are  such  as  correspond  to  the  time  of  their 
rotation. 

The  rotation  of  the  earth  gives  to  the  celestial  sphere  the  ap- 
pearance of  revolving  in  the  contrary  direction,  as  the  forward 
motion  of  a  boat  on  a  river  gives  to  the  banks  an  appearance  of 
backward  motion ;  and  since  the  apparent  motion  of  the  heavens  is 
from  east  to  west,  the  real  rotation  of  the  earth  which  produces  that 
appearance  must  be  from  west  to  east. 

24.  The  earth's  axis  is  the  diameter  around  which  it  revolves 
once  a  day.    The  extremities  of  this  axis  are  the  terrestrial  poles; 
one  is  called  the  north  pole,  and  the  other  the  south  pole. 

The  terrestrial  equator  is  a  great  circle  of  the  earth  perpendicu- 
lar to  the  earth's  axis. 

Meridians  are  great  circles  passing  through  the  poles  of  the 
earth. 

25.  The  latitude  of  a  place  is  the  arc  of  the  meridian  which  ia 
comprehended  between  that  place  and  the  equator.     Latitude  is 
reckoned  north  and  south  of  the  equator,  from  0  to  90°. 

A  parallel  of  latitude  is  any  small  circle  on  the  earth's  surface 
parallel  to  the  terrestrial  equator.  These  parallels  continually 
diminish  in  size  as  we  proceed  from  the  equator  to  the  pole. 

The  polar  distance  of  a  place  is  its  distance  from  the  nearest 
pole,  and  is  the  complement  of  the  latitude. 

The  longitude  of  a  place  is  the  arc  of  the  equator  intercepted 
between  the  meridian  of  that  place  and  some  assumed  meridian 
to  which  all  others  are  referred.  The  English  reckon  longitude 
from  the  observatory  of  Greenwich,  the  French  from  the  observa- 
tory of  Paris,  and  the  Germans  from  the  observatory  of  Berlin, 
or  from  the  island  of  Ferro,  which  is  assumed  to  be  20°  west  of 


FIGURE   AND   DIMENSIONS   OF  THE   EARTH. 


19 


the  observatory  in  Paris.  In  the  United  States  we  sometimes 
reckon  longitude  from  Washington,  and  sometimes  from  Green- 
wich. Longitude  is  usually  reckoned  :east  and  west  of  the  first 
meridian,  from  0  to  180°.  The  longitude  and  latitude  of  a  place 
determine  its  position  on  the  earth's  surface. 

26.  The  latitude  of  a  place.— Let  SJENQ  represent  the  earth 
surrounded  by  the  distant  starry  sphere  HZOK.  The  diameter 
of  the  earth  being  insignificant  in  comparison  with  the  distance 
of  the  stars,  the  appearance  of  the  heavens  will  be  the  same 


Fig.  C. 


whether  they  are  viewed  from  the  centre  of  the  earth,  or  from 
any  point  on  its  surface.  Suppose  the  observer  to  be  at  P,  a 
point  on  the  surface  between  the  equator  M  and  the  north  pole 
N.  The  latitude  of  this  place  is  JEP,  or  the  angle  ^ECP.  If  the 
line  PC  be  continued  to  the  firmament,  it  will  pass  through  the 
point  Z,  which  is  the  zenith  of  the  observer.  If  the  terrestrial 
axis  KS  be  continued  to  the  firmament,  it  will  pass  through  the 
celestial  poles  N'  and  S'.  If  the  terrestrial  equator  ^EQ  be  con- 
tinued to  the  heavens,  it  will  constitute  the  celestial  equator 
.^E'Q'.  The  observer  at  P  will  see  the  entire  hemisphere  HZO, 
of  which  his  zenith  Z  is  the  pole.  The  other  hemisphere  will  be 
concealed  by  the  earth. 

The  arc  N'O -contains  the  same  number  of  degrees  as  ^E'Z,  and 


20  ASTRONOMY. 

the  arc  ZW  is  the  complement  of  ON';  that  is,  the  altitude  of  the 
visible  pok  is  equal  to  the  latitude  of  the  place,  and  the  zenith  dis- 
tance of  the  visible  pole  is  the  complement  of  the  latitude. 

27.  How  the  latitude  of  a  place  may  be  determined. — If  there  were 
a  star  situated  precisely  at  the  pole,  its  altitude  would  be  the  lati- 
tude of  the  place.     The  pole  star  describes  a  small  circle  around 
the  pole,  and  crosses  the  meridian  twice  in  each  revolution,  once 
above  and  once  below  the  pole.     The  half  sum  of  the  altitudes  in 
these  two  positions  is  equal  to  the  altitude  of  the  pole ;  that  is, 
to  the  latitude  of  the  place.     The  same  result  would  be  obtained 
by  observing  any  circumpokr  star  on  the  meridian  both  above 
and  below  the  pole. 

28.  Circles  which  pass  through  the  two  poles  of  the  celestial 
sphere  are  called  hour  circles.    If  two  such  circles  include  an  arc 
of  15°  of  the  celestial  equator,  the  interval  between  the  instants 
of  their  coincidence  with  the  meridian  will  be  one  hour. 

29.  The  right  ascension  of  a  star  is  the  arc  of  the  celestial  equa- 
tor comprehended  between  a  certain  point  on  the  equator  called 
the  first  point  of  Aries,  and  an  hour  circle  passing  through  that 
star.     Right  ascension  is  sometimes  expressed  in  degrees,  min- 
utes, and  seconds  of  arc,  but  generally  in  hours,  minutes,  and  sec- 
onds of  time.     It  is  reckoned  eastward  from  zero  up  to  2-1  hours, 
or  360  degrees.    If  the  hands  of  the  sidereal  clock  be  set  to 
Oh.  Om.  Os.  when  the  first  point  of  Aries  is  on  the  meridian,  the 
clock  (if  it  neither  gains  nor  loses  time)  will  afterward  indicate  at 
each  instant  the  right  ascension  of  any  object  which  is  then  on 
the  meridian,  for  the  motion  of  the  hands  of  the  clock  corresponds 
exactly  with  the  apparent  diurnal  motion  of  the  heavens.    While 
15°  of  the  equator  pass  the  meridian,  the  hands  of  the  clock  move 
through  one  hour. 

The  sidereal  day  therefore  begins  when  the  first  point  of  Aries 
crosses  the  meridian,  and  the  sidereal  clock  should  always  indi- 
cate Oh.  Om.  Os.  when  the  first  of  Aries  is  on  the  meridian. 

30.  The  distance  of  an  object  from  the  celestial  equator,  meas- 
ured upon  the  hour  circle  which  passes  through  it,  is  called  its 
declination,  and  is  north  or  south  according  as  the  object  is  on 


FIGURE   AND  DIMENSIONS   OF  THE  EARTH.  21 

the  north  or  south  side  of  the  equator.  North  declination  is  in- 
dicated by  the  sign  +  ,  and  south  declination  by  the  sign  — . 

The  position  of  an  object  on  the  firmament  is  indicated  by  its 
declination  and  right  ascension.  Its  declination  expresses  its  dis- 
tance north  or  south  of  the  celestial  equator,  ancj  its  right  ascen- 
sion expresses  the  distance  of  the  hour  circle  upon  which  it  is  sit- 
uated, from  a  fixed  point  upon  the  celestial  equator. 

The  north  polar  distance  of  a  star  is  its  distance  from  the  north 
pole. 

31.  A  right  sphere. — The  celestial  sphere  presents  different  ap- 
pearances to  observers  in  different  latitudes.     If  the  observer 
were  situated  at  the  terrestrial  equator,  the  poles  would  lie  in  the 
horizon,  the  celestial  equator  would  be  perpendicular  to  the  plane 
of  the  horizon,  and  hence  the  horizon  would  bisect  the  equator 
and  all  circles  parallel  to  it.    Therefore  all  celestial  objects  would 
be  for  equal  periods  above  and  below  the  horizon,  and  they  would 
appear  to  rise  perpendicularly  on  the  eastern  side  of  the  horizon, 
and  set  perpendicularly  on  the  western  side.     Such  a  sphere  is 
called  a  right  sphere,  the  diurnal  motion  being  at  right  angles  to 
the  horizon. 

32.  A  parallel  sphere. — At  one  of  the  poles  of  the  earth,  the  ce- 
lestial pole  being  in  the  zenith,  the  celestial  equator  would  coin- 
cide with  the  horizon,  and  by  the  diurnal  motion  all  celestial  ob- 
jects would  move  in  circles  parallel  to  the  horizon.     This  is 
called  a  parallel  sphere.     In  a  parallel  sphere,  an  object  upon 
the  equator  will  be  carried  by  the  diurnal  motion  round  the  hori- 
zon, without  either  rising  or  setting. 

33.  An  oblique  sphere. — At  all  latitudes  between  the  equator  and 
the  pole,  the  celestial  equator  is  inclined  to  the  horizon  at  an  angle 
equal  to  the  distance  of  the  pole  from  the  zenith  ;*  that  is,  equal 
to  the  complement  of  the  latitude.    The  parallels  DF,  GK,  Fig.  6, 
are  unequally  divided  by  the  horizon ;  that  is,  all  objects  between 
the  celestial  equator  and  the  visible  pole  are  longer  above  than 
below  the  horizon,  and  all  objects  on  the  other  side  of  the  equa- 
tor are  longer  below  than  above  the  horizon. 

A  parallel,  BO,  whose  distance  from  the  visible  pole  is  equal  to 
the  latitude,  is  entirely  above  the  horizon ;  and  the  same  is  true 


22  ASTRONOMY. 

of  all  parallels  still  nearer  to  that  pole.  Also  the  parallel  HL, 
whose  distance  from  the  invisible  pole  is  equal  to  the  latitude,  is 
entirely  below  the  horizon ;  and  the  same  is  true  of  all  parallels 
still  nearer  to  that  pole.  Hence,  in  the  United  States,  stars  within 
a  certain  distance  of  the  north  pole  never  set,  and  stars  at  an 
equal  distance  from  the  south  pole  never  rise. 

The  circle  BO  is  called  the  circle  of  perpetual  apparition,  be- 
cause the  stars  which  are  included  within  it  never  set.  The  ra- 
dius of  this  circle  is  equal  to  the  latitude  of  the  place. 

The  circle  HL  is  called  the  circle  of  perpetual  occullation,  be- 
cause the  stars  which  are  included  within  it  never  rise.  The  ra- 
dius of  this  circle  is  also  equal  to  the  latitude  of  the  place. 

The  celestial  sphere  here  described  is  called  an  oblique  sphere, 
the  diurnal  motion  being  oblique  to  the  horizon. 

Whether  the  sphere  be  right  or  oblique,  one  half  of  the  celes- 
tial equator  will  be  below  the  horizon,  and  the  other  half  above 
it.  Every  object  on  the  equator  will  therefore  be  above  the  ho- 
rizon during  as  long  a  time  as  it  is  below,  and  will  rise  and  set 
at  the  east  and  west  points. 

34.  Effects  of  centrifugal  force. — We  have  discovered  that  the 
earth  has  a  globular  figure,  and  that  it  rotates  upon  its  axis  once 
in  24  sidereal  hours.  But,  since  the  earth  rotates  upon  an  axis, 
its  form  can  not  be  thai  of  a  perfect  sphere;  for  every  body  revolv- 
ing in  a  circle  acquires  a  centrifugal  force  which  tends  to  make  it 
recede  from  the  centre  of  the  circle.  Every  particle,  P,  upon  the 
earth's  surface  acquires,  therefore,  a  force  which  acts  in  a  direc- 
Fig  T.  N  tion,  EP,  perpendicular  to  the  axis 

of  rotation.     This  centrifugal  force, 
which  we  will  represent  by  PA,  may 
be  resolved  into  two  other  forces  PB 
and  PD,  one  acting  in  the  direction 
of  a  radius  of  the  earth,  and  the  other 
at  right  angles  to  the  radius.     The 
former,  being  opposed  to  the  earth's 
attraction,  has  the  effect  of  diminish- 
ing the  weight  of  the  body ;  the  latter,  being  directed  toward  the 
equator,  tends  to  produce  motion  in  the  direction  of  the  equator. 
The  intensity  of  the  centrifugal  force  increases  with  the  radius 
of  the  circle  described,  and  is  therefore  greatest  at  the  equator. 


FIGURE  AXD  DIMENSIONS  .OF  THE   EARTH.  23 

Moreover,  the  nearer  the  point  is  to  the  equator,  the  more  direct- 

ly is  the  centrifugal  force  opposed  to  the  weight  of  the  body. 
The  effects,  therefore,  produced  by  the  rotation  of  the  earth  are, 
1st.  All  bodies  decrease  in  weight  in  going  from  the  pole  to  the 

equator;  and, 

2d.  All  bodies  which  are  free  to  move,  tend  from  the  higher 

latitudes  toward  the  equator. 

35.  The  effect  of  centrifugal  force  computed.  —  Let  A  be  a  ball  at- 
tached to  a  string  AS  ;  let  S  be  a  fixed  point,  and  ACE  the  circle 
in  which  the  ball  revolves,  and  AC  the  arc  Fig.  s. 

which  the  ball  describes  in  a  given  time.    When        ±^E^r~~~\ 
the  ball  was  at  A,  it  was  moving  in  the  direc-     /  \ 

tion  of  the  tangent  AB,  and  it  would  continue 
in  this  direction  if  it  were  acted  upon  by  no 
other  force  than  the  first  impulse  ;  but  we  find  V  \ 
it  deflected  into  the  diagonal  AC,  and  this  diag-  \^  \ 
onal  is  the  resultant  of  two  forces  represented 
by  AB,  AD.  Now  AB  represents  the  path  which  the  ball  would 
describe  under  the  first  impulse,  and  therefore  AD  represents  the 
motion  impressed  upon  it  by  the  tension  of  the  string,  and  which 
deflects  the  ball  from  the  tangent  to  the  circle. 

AP2 
AD  :  AC  ::  AC  :  AE;  whence  AD  =  £f-. 

^xi.O 

If  Y  represent  the  velocity  of  the  revolving  body  expressed  in 
feet  per  second,  and  R  the  length  of  the  string  in  feet,  we  shall 
have  the  deflection  from  a  tanent  line 


36.  Centrifugal  force  compared  with  the  force  of  gravity.  —  "We 
may  compare  the  centrifugal  force  of  a  body  with  the  force  of 
gravity,  by  comparing  the  spaces  through  which  the  body  would 
move  in  a  given  time,  under  the  operation  of  these  two  forces. 

Let  W=  the  weight  of  the  revolving  body,  C  =  its  centrifugal 
force,  and  #=16  feet,  the  space  through  which  W  would  fall  free- 
l in  one  second.  Then  we  shall  have 


W.V2 

whence  C=     '     . 

2% 


24  ASTRONOMY. 

We  may  also  express  the  centrifugal  force  of  a  revolving  body 
by  reference  to  the  number  of  revolutions  made  iii  a  given  time. 
Let  N  represent  the  number  of  revolutions,  or  the  fraction  of  a 
revolution  performed  by  the  body  in  one  second.  The  circum- 
ference of  the  circle  which  the  body  describes  will  be  27rR  The 
space  through  which  the  body  moves  in  one  second,  that  is,  its 
velocity,  is  27J-K.N.  Hence  we  have 

w£wr=M  x  RN,.W=1.2275  x  Rmw. 


The  amount  of  the  loss  of  weight  produced  at  the  equator  by 
centrifugal  force,  may  be  computed  as  follows  : 

The  radius  of  the  equator  is  20,923,600  feet;  and  since  the 
time  of  one  rotation  is  23h.  56m.  4s.,  or  86164  seconds,  N  =  -5^^ 

Hence  0  =  1.2275  x  20,923,600  x  r,  *  W, 


Thus  we  find  that  at  the  equator  the  centrifugal  force  of  a  body 
arising  from  the  earth's  rotation,  is  -^y  part  of  the  weight;  and 
since  this  force  is  directly  opposed  to  gravity,  the  weight  must 
sustain  a  loss  of  ir|ir  part. 

37.  Centrifugal  force  at  any  latitude.  —  The  centrifugal  force  at 
the  equator  is  to  the  centrifugal  force  in  any  other  latitude  as 
radius  to  the  cosine  of  \he  latitude.     But  the  entire  centrifugal 
force  at  any  latitude  is  to  that  part  of  the  centrifugal  force  which 
is  opposed  to  the  weight  of  the  body,  as  radius  to  the  cosine  of 
the  latitude  ;  that  is,  the  loss  of  weight  of  a  body  caused  by  the 
centrifugal  force  at  any  latitude,  is  -^  of  the  weight  multiplied 
by  the  square  of  the  cosine  of  the  latitude. 

38.  Effect  of  centrifugal  force  t/^on  the  form  of  a  body.  —  A  por- 
tion, PD,  of  the  centrifugal  force  causes  a  tendency  to  move  to- 
ward the  equator.    If  the  surface  of  the  globe  were  entirely  solid, 
this  tendency  would  be  counteracted  by  the  cohesion  of  the  par- 
ticles.   But  since  a  portion  of  the  earth's  surface  is  fluid,  this 
portion  must  yield  to  the  centrifugal  force,  and  flow  toward  the 
equator.     Thus  the  water  must  recede  from  the  higher  latitudes 
in  either  hemisphere,  and  accumulate  around  the  equator.     The 
earth,  therefore,  instead  of  being  an  exact  sphere,  must  become 


FIGURE  AND  DIMENSIONS   OF  THE   EARTH. 


25 


an  oblate  spheroid.  A  globe  consisting  of  any  plastic  material 
would  be  reduced  to  such  a  figure  by  causing  it  to  rotate  rapidly 
upon  'an  axis.  The  amount  of  the  elliptic!  ty  of  the  earth  must 
depend  upon  the  centrifugal  force,  and  the  attraction  exerted  by 
the  earth  upon  bodies  placed  on  its  surface. 

39.  Weight,  of  a  body  at  the2)ole  and  the  equator.  —  "We  have  found 
that  at  the  equator  the  loss  of  weight  due  to  centrifugal  force  is 
•2-j-g-.     From  a  comparison  of  observations  of  the  length  of  the 
seconds'  pendulum  made  in  different  parts  of  the  globe,  it  is 
found  that  the  weight  of  a  body  at  the  pole  actually  exceeds  its 
weight  at  the  equator,  by  -5-^-5-. 

The  difference  between  these  fractions  is  TTT  —  Trg-^  =  Ts-u  5  that 
is,  the  actual  attraction  exerted  by  the  earth  upon  a  body  at  the 
equator  is  less  than  at  the  pole,  by  the  590th  part  of  the  whole 
weight.  This  difference  is  due  to  the  elliptic  form  of  the  merid- 
ians, by  which  the  distance  of  the  body  at  the  equator  from  the 
centre  of  the  earth  is  increased. 

40.  How  an  arc  of  a  meridian  is  measured.  —  Numerous  arcs  of 
the  meridian  have  been  measured,  for  the  purpose  of  accurately 
determining  the  figure  and  dimensions  of  the 

earth.     These  arcs  are  measured  in  the  fol- 
lowing manner: 

A  level  spot  of  ground  is  selected,  where  a 
base  line,  AB,  from  five  to  ten  miles  in  length,  c 
is  measured  with  the  utmost  precision.  A 
third  station,  C,  is  selected,  forming  with  the 
base  line  a  triangle  as  nearly  equilateral  as  is 
convenient.  The  angles  of  this  triangle  are 
measured  with  a  theodolite,  and  the  two  re- 
maining sides  may  then  be  computed.  A 
fourth  station,  D,  is  now  selected,  forming  with 
two  of  the  former  stations  a  second  triangle, 
in  which  all  the  angles  are  measured;  and 
since  one  side  is  already  known,  the  others 
may  be  computed.  A  fifth  station,  E,  is  then 
selected,  forming  a  third  triangle;  and  thus 
we  proceed  forming  a  series  of  triangles,  fol- 
lowing nearly  the  direction  of  a  meridian. 


rig.  9. 


26  ASTRONOMY. 

Having  determined  by  astronomical  observations  the  azimuth 
of  one  of  the  lines,  AB,  that  is,  its  inclination  to  a  meridian  line, 
we  can  compute  how  much  any  station,  A,  is  north  of  any  other 
station,  II.  The  latitude  of  each  of  these  stations  being  deter- 
mined, we  know  the  distance  in  miles  which  corresponds  to  a 
given  difference  of  latitude,  and  hence,  by  a  proportion,  we  can 
find  the  length  of  an  arc  of  one  degree.  This  method  of  survey- 
ing is  susceptible  of  great  precision,  because  we  may  choose  the 
most  favorable  site  for  measuring  a  base  line;  and  after  this 
nothing  more  is  required  but  the  measurement  of  angles,  which, 
with  a  large  theodolite,  can  be  done  with  extreme  accuracy. 

41.  Verification  of  the  icork. — In  order  to  verify  the  entire  work, 
a  second  base  line  is  measured  near  the  end  of  the  series  of  tri- 
angles, and  we  compare  its  measured  length  with  the  length  as 
computed  from  the  first  base,  through  the  intervention  of  the 
series  of  triangles.    In  the  survey  of  the  coast  of  the  United  States, 
three  base  lines  have  been  measured  east  of  New  York,  the  short- 
est being  a  little  more  than  five  miles  in  length,  and  the  longest 
more  than  ten  miles,  and  the  two  extreme  bases  are  distant  from 
each  other  430  miles  in  a  direct  line.     In  one  instance,  the  ob- 
served length  of  a  base  differs  from  its  length  deduced  from  one  of 
the  other  bases  by  six  inches;  in  the  other  cases  the  discrepancy 
does  not  exceed  three  inches.     This  coincidence  proves  that  none 
but  errors  of  extreme  minuteness  have  been  committed  in  the 
determination  of  the  position  of  the  intermediate  stations  stretch- 
ing from  the  city  of  New  York  to  the  eastern  boundary  of  Maine. 

42.  Results  of  measurements.  —  In  the  manner  here  described, 
arcs  of  a  meridian  have  been  measured  in  nearly  every  country 
of  Europe.     These  surveys  form  a  connected  chain  of  triangles, 
extending  from  the  North  Cape,  in  lat.  70°  40',  to  an  island  in  the 
Mediterranean  in  lat.  38°  42';  and  a  survey  is  now  in  progress 
extending  across  the  Mediterranean  through  Algeria,  and  is  to  be 
continued  to  the  Sahara.     An  arc  has  been  measured  in  India 
extending  from  lat.  29°  26'  to  lat  8°  5'.     An  arc  has  been  meas- 
ured in  South  America  extending  from  the  equator  to  more  than 
three  degrees  of  south  latitude.     An  arc  of  over  four  degrees  has 
also  been  measured  in  South  Africa.    The  survey  of  the  eoajst  of 
the  United  States  will  ultimately  furnish  several  important  arcs 
of  a  meridian,  but  these  observations  are  not  yet  fully  reduced. 


FIGURE  AND  DIMENSIONS  OF  THE   EARTH.  27 

These  measurements  enable  us  to  determine  with  great  accuracy 
the  length  of  a  degree  of  latitude  for  the.  entire  distance  from  the 
equator  to  the  north  pole.  The  results  are 

A  degree  at  the  equator  =68.702  miles,  or  362,748  feet. 
"      inlat.  45°        =69.048      "      "364,572     " 
"      at  the  pole      =69.396      "      "366,410    " 
A  degree  at  the  pole       )  _  u      u 

minus  a  degree  at  equator  f  =  3'662 

43.  Conclusion  from  these  results. — If  the  earth  were  perfectly 
spherical,  a  terrestrial  meridian  would  be  an  exact  circle,  and  ev- 
ery part  of  it  would  have  the  same  curvature ;  that  is,  a  degree 
of  latitude  would  be  every  where  the  same.  But  we  have  found 
that  the  length  of  a  degree  increases  as  we  proceed  from  the 
equator  toward  the  poles,  and  the  amount  of  this  difference  af- 
fords a  measure  of  the  departure  of  a  meridian  from  the  figure  of 
a  circle. 

The  plumb-line  must  every  where  be  perpendicular  to  the  sur- 
face of  tranquil  water,  and  can  not,  therefore,  every  where  point 
exactly  toward  the  earth's  centre.  Let  A,  B  be  two  plumb-lines 
suspended  on  the  same  meridian  near 
the  equator,  and  at  such  a  distance 
from  each  other  as  to  be  inclined  at 
an  angle  of  1°.  Let  C  and  D  be  two 
other  plumb-lines  on  a  meridian  near 
one  of  the  poles,  also  making  with 
each  other  an  angle  of  1°.  The  dis- 
tance from  A  to  B  is  found  to  be  less  than  from  C  to  D,  from 
which  we  conclude  that  the  meridian  curves  more  rapidly  near 
A  than  near  C. 

It  is  found  that  all  the  observations  in  every  part  of  the  world 
are  very  accurately  represented  by  supposing  the  meridian  to  be 
an  ellipse,  of  which  the  polar  diameter  is  the  minor  axis. 

The  equatorial  diameter  of  this  ellipse  is  7926.708  miles, 
the  polar  diameter         "         "          "       7899.755     " 
the  difference  is  26.953     " 

That  is,  the  equatorial  diameter  exceeds  the  polar  diameter  by 
-jl^th  of  its  length.  This  difference  is  called  the  ellipticity  of  the 
earth. 

The  meridional  circumference  of  the  earth  is  24,857.5  miles. 


>8  ASTRONOMY. 

From  measurements  which  have  been  made  at  right  angles  tc 
the  meridian,  it  appears  that  the  equator  and  parallels  of  latitude 
are  very  nearly,  if  not  exactly,  circles.  Hence  it  appears  that  the 
form  of  the  earth  is  that  of  an  oblate  spheroid;  which  is  a  solid 
generated  by  the  revolution  of  a  semi-ellipse  about  its  minor  axis. 

44.  Loss  of  weight  at  the  equator  explained. — It  has  been  mathe- 
matically proved  that  a  spheroid  whose  ellipticity  is  ^^7,  and 
whose  mean  density  is  double  the  density  at  the  surface,  exerts 
an  attraction  upon  a  particle  placed  at  its  pole,  greater  by  -s-g-uth. 
part  than  the  attraction  upon  a  particle  at  its  equator ;  and  this 
we  have  seen  is  the  fraction  which  must  be  added  to  the  loss  of 
weight  by  centrifugal  force,  to  make  up  the  total  loss  of  weight  at 
the  equator,  as  shown  by  experiments  with  the  seconds'  pendulum. 

This  coincidence  may  be  regarded  as  demonstrating  that  the 
earth  does  rotate  upon  its  axis  once  in  24  hours. 

45.  Equatorial  protuberance. — If  a  sphere  be  conceived  to  be  in- 
rig.  it  ^^*r^^  scribed  within  the  terrestrial  spheroid,  hav- 
ing the  polar  axis  NS  for  its  diameter,  a 
spheroidal  shell  will  be  included  between  its 

)D  surface  and  that  of  the  spheroid,  having  a 
thickness,  AB,  of  13  miles  at  the  equator,  and 
becoming  gradually  thinner  toward  the  poles. 
This  shell  of  protuberant  matter,  by  means 

of  its  attraction,  gives  rise  to  many  important  phenomena,  as  will 

be  explained  hereafter. 

Tlie  Density  of  tie  Earth. 

46.  Three  methods  have  been  practiced  for  determining  the 
average  density  of  the  earth.     These  methods  are  all  founded 
upon  the  principle  of  comparing  the  attraction  which  the  earth 
exerts  upon  any  object,  with  the  attraction  which  some  other 

-body,  whose  mass  is  known,  exerts  upon  the  same  object. 

First  metiiod.—Ry  comparing  the  attraction  of  the  earth  with 
that  of  a  small  mountain. 

^  In  1714,  Dr.Maskelyne  determined  the  ratio  of  the  mean  den- 
pity  of  the  earth  to  that  of  a  mountain  in  Scotland,  called  Sche- 
hallien,  by  ascertaining -how  much  the  local  attraction  of  the 
fountain  deflected  a  plumb-line  from  a  vertical  position.  This 


THE  DENSITY  OF  THE  EARTH. 


29 


Fig.  12. 


mountain  stands  alone  on  an  extensive  plain,  so  that  there  are  no 
neighboring  eminences  to  affect  the  plumb-line.  Two  stations 
were  selected,  one  on  its  northern  and  the  other  on  its  southern 
side,  and  both  nearly  in  the  same  meridian.  A  plumb-line,  at- 
tached to  an  instrument  called  a  zenith 
sector,  designed  for  .measuring  small  ze- 
nith distances,  was  set  up  at  each  of  these 
stations,  and  the  distance  from  the  direc- 
tion of  the  plumb-line  to  a  certain  star 
was  measured  at  each  station,  the  instant 
that  the  star  was  on  the  meridian.  The 
difference  between  these  distances  gave 
the  angle  formed  by  the  two  directions 
of  the  plumb-lines  AE,  CG.  Were  it  not 
for  the  mountain,  the  plumb-lines  would  take  the  positions  AB, 
CD ;  and  the  angle  which  they  would,  in  that  case,  form  with  each 
other,  is  found  by  measuring  the  distance  between  the  two  sta- 
tions, and  allowing  about  one  second  for  every  hundred  feet. 

In  Dr.  Maskelyne's  experiment,  the  distance  between  the  two 
stations  was  4000  feet;  so  that  if  the  direction  of  gravity  had  not 
been  influenced  by  the  mountain,  the  inclination  of  the  plumb- 
lines  at  the  two  places  would  have  been  41  seconds.  The  inclina- 
tion was  actually  found  to  be  55".  The  difference,  or  12",  is  to 
be  ascribed  to  the  attraction  of  the  mountain.  It  was  computed 
that  if  the  mountain  had  been  as  dense  as  the  interior  of  the  earth, 
the  disturbance  would  have  been  about  21".  Therefore,  the  ra- 
tio of  the  density  of  the  mountain  to  that  of  the  entire  earth,  was 
that  of  12  to  21. 

The  mean  density  of  the  mountain  was  ascertained  by  numer- 
ous borings  to  be  2.75  times  that  of  water.  Hence  the  mean 
density  of  the  earth  was  concluded  to  be  4.95  times  that  of 
water. 

In  the  year  1855,  observations  were  made  for  ascertaining  the 
deviation  of  the  plumb-line  produced  by  the  attraction  of  Arthur's 
Seat,  a  hill  822  feet  high,  near  Edinburg,  from  which  the  mean 
density  of  the  earth  was  computed  to  be  5.32. 

47.  Second  method. — The  mean  density  of  the  earth  has  been 
determined  by  experiments  with  the  torsion  balance. 

In  the  year  1798,  Cavendish  compared  the  attraction  of  the 


30  ASTRONOMY. 

earth  with  the  attraction  of  two  lead  balls,  each  of  which  was  one 
foot  in  diameter.  The  bodies  upon  which  their  attraction  was 
exerted  were  two  leaden  balls,  each  about  two  inches  in  diameter. 
They  were  attached  to  the  ends  of  a  slender  wooden  rod  six  feet 
in  length,  which  was  supported  at  the  centre  by  a  fine  wire  40 
inches°long.  The  balls,  if  left  to  themselves,  will  come  to  rest 
when  the  supporting  wire  is  entirely  free  from  torsion,  but  a  very 
slight  force  is  sufficient  to  turn  it  out  of  this  plane.  The  position 
of°he  supporting  rod  was  accurately  observed  with  a  fixed  tel- 
escope. The  large  balls  were  then  brought  near  the  small  ones, 
but  on  opposite  sides,  so  that  the  attraction  of  both  balls  might 
conspire  to  twist  the  wire  in  the  same  direction,  when  it  was 
found  that  the  small  balls  were  sensibly  attracted  by  the  larger 
ones,  and  the  amount  of  this  deflection  was  carefully  measured. 
The  large  balls  were  then  moved  to  the  other  side  of  the  small 
ones,  when  the  rod  was  found  to  be  deflected  in  the  contrary  di- 
rection, and  the  amount  of  this  deflection  was  recorded.  This 
experiment  was  repeated  seventeen  times. 

These  experiments  furnish  a  measure  of  the  attraction  of  the 
large  balls  for  the  small  ones,  and  hence  we  can  compute  what 
would  be  their  attraction  if  they  were  as  large  as  the  earth.  But 
we  know  the  attraction  actually  exerted  by  the  earth  upon  the 
Email  balls,  it  being  measured  by  the  weight  of  the  balls.  Thus 
we  know  the  attractive  force  of  the  earth  compared  with  that  of 
the  lead  balls;  and  since  we  know  the  density  of  the  lead,  we 
can  compute  the  average  density  of  the  earth.  From  these  ex- 
periments, Cavendish  concluded  that  the  mean  density  of  the 
c%arth  was  5.45. 

These  experiments  were  repeated  by  Dr.  Reich,  at  Frcyberg, 
in  Saxony,  in  the  year  1836,  and  the  mean  of  57  trials  gave  a  re- 
sult of  5.44. 

In  the  years  1841-'2  a  similar  series  of  experiments  was  con- 
ducted with  the  greatest  care  by  Sir  Francis  Baily  in  England, 
and  from  over  2000  trials  he  concluded  the  mean  density  of  the 
earth  to  be  5.67. 

48.  Third  method. — The  mean  density  of  the  earth  may  be  de- 
termined by  means  of  pendulum  experiments  at  the  top  and  bot- 
tom of  a  deep  mine.  The  rate  of  vibration  of  a  pendulum  der 
pends  upon  the  intensity  of  the  earth's  attraction,  and  thus  be- 


THE  DENSITY  OF  THE  EARTH.  31 

comes  a  measure  of  this  intensity.  If  we  vibrate  the  same  pen- 
dulum at  the  top  and  bottom  of  a  mine  whose  depth  is  1000  feet, 
we  shall  have,  a  measure  of  the  force  of  gravity  at  the  bottom  of 
the  mine  compared  with  the  force  at  the  top.  Now  at  the  top  of 
the  mine  the  pendulum  is  attracted  by  every  particle  of  matter  in 
the  globe ;  but,  since  a  spherical  shell  may  be  shown  to  exert  no 
influence  upon  a  point  situated  within  it,  the  pendulum  at  the 
bottom  of  the  mine  will  only  be  influenced  by  a  sphere  whose  ra- 
dius is  1000  feet  less  than  that  of  the  earth.  We  thus  obtain  the 
attraction  of  this  external  shell,  whose  thickness  is  1000  feet,  com- 
pared with  the  attraction  of  the  entire  globe;  and  since  the  vol- 
umes of  both  these  bodies  may  be  computed,  we  are  able  to  de- 
duce the  average  density  of  the  globe,  compared  with  that  of  the 
external  shell.  Now,  by  actual  examination,  we  can  determine 
the  density  of  the  strata  penetrated  by  the  mine,  and  hence  we 
are  able  to  compute  the  mean  density  of  the  globe. 

This  method  was  applied  in  one  of  the  mines  of  England,  near 
Newcastle,  in  the  year  1854.  The  depth  of  the  mine  was  1256 
feet;  and  it  was  found  that  a  pendulum  which  vibrated  seconds 
at  the  top  of  the  mine,  when  transferred  to  the  bottom  of  the 
mine  gained  2^  seconds  per  day.  From  this  it  was  computed 
that  the  force  of  gravity  at  the  bottom  of  the  mine  was  15^8C 
greater  than  at  the  top  of  the  mine ;  and  hence  it  was  computed 
that  the  average  density  of  the  globe  was  2.62  times  that  of  the 
external  shell.  By  actual  examination,  it  was  found  that  the  av- 
erage density  of  the  rocks  penetrated  by  the  mine  was  2.5,  whence 
it  follows  that  the  mean  density  of  the  earth  is  6.56. 

49.  Fourth  method. — In  a  somewhat  similar  manner,  we  may 
determine  the  density  of  the  earth  by  comparing  the  length  of 
the  pendulum  vibrating  seconds  on  the  summit  of  a  mountain, 
with  that  at  the  base  of  the  mountain.  In  1824,  the  vibrations 
of  a  pendulum  on  Mount  Cenis,  in  Italy,  at  an  elevation  of  6734 
English  feet,  were  compared  with  the  vibrations  near  the  level  of 
the  sea,  and  the  density  of  the  earth  was  hence  deduced  to  be  4.84. 

The  average  of  these  seven  determinations  is  5.46,  which  must 
be  a  tolerable  approximation  to  the  truth. 

These  results  verify,  in  a  remarkable  manner,  the  conjecture  of 
Newton,  who,  in  1680,  estimated  that  the  average  density  of  the 
earth  was  5  or  6  times  greater  than  that  of  water. 


32  ASTRONOMY. 

50.  Volume  and  weight  of  die  earth.— Having  determined  the  di- 
mensions of  the  earth,  we  can  easily  compute  its  volume,  and, 
knowing  its  density,  we  can  also  compute  its  weight.    Its  volume 
is  found  to  contain 

259,400  millions  of  cubic  miles. 

Also  the  total  weight  of  the  earth  is  6  sextillions  of  tons — a 
number  expressed  by  the  figure  6  with  21  ciphers  annexed. 

51.  Direct  proof  of  the  earth's  rotation. — A  direct  proof  of  the 
earth's  rotation  is  derived  from  observations  of  a  pendulum.     If 
a  heavy  ball  be  suspended  by  a  flexible  wire  from  a  fixed  point, 
and  the  pendulum  thus  formed  be  made  to  vibrate,  its  vibrations 
will  all  be  performed  in  the  same  plane.     If,  instead  of  being  sus- 
pended from  a  fixed  point,  we  give  to  the  point  of  support  a  slow 
movement  of  rotation  around  a  vertical  axis,  the  plane  of  vibra- 
tion will  still  remain  unchanged.     This  may  be  proved  by  hold- 
ing in  the  fingers  a  pendulum  composed  of  a  simple  ball  and 
string,  and  causing  it  to  vibrate.     Upon  twirling  the  string  be- 
tween the  fingers,  the  ball  will  be  seen  to  rotate  on  its  axis,  with- 
out, however,  changing  its  plane  of  vibration. 

Suppose,  then,  a  heavy  ball  to  be  suspended  by  a  wire  from  a 
fixed  point  directly  over  the  pole  of  the  earth,  and  made  to  vi- 
brate ;  these  vibrations  will  continue  to  be  made  in  the  same  in- 
variable plane.  But  the  earth  meanwhile  turns  round  at  the  rate 
of  15°  per  hour;  and  since  the  observer  is  unconscious  of  his  own 
motion  of  rotation,  it  results  that  the  plane  of  vibration  of  the 
pendulum  appears  to  revolve  at  the  same  rate  in  the  opposite  di- 
rectidn. 

If  the  pendulum  be  removed  to  the  equator,  and  set  vibrating 
in  the  direction  of  a  meridian,  the  plane  of  vibration  will  still  re- 
main unchanged;  and  since,  notwithstanding  the  earth's  rotation, 
this  plane  always  coincides  with  a  meridian,  the  plane  of  vibra- 
tion appears  to  remain  unchanged. 

52.  Phenomena  in  the  middle  latitudes. — At  places  intermediate 
between  the  pole  and  the  equator,  the  apparent  motion  of  the 
plane  of  vibration  is  less  than  15°  per  hour,  and  diminishes  as  we 
recede  from  the  poK    This  may  be  proved  in  the  following  man- 
ner: 


PEOOF  OF  THE  EARTH'S  ROTATION. 


33 


Let  NPSE  represent  a  meridian  of 
the  earth,  AP  a  tangent  to  this  circle 
at  P,  meeting  the  earth's  axis  produced 
in  A.  Suppose  a  pendulum  to  be  set 
up  at  the  point  P,  and  vibrated  in  the 
plane  of  the  meridian.  When  by  the 
rotation  of  the  earth  the  point  P  is 
brought  to  P',  the  plane  of  vibration 
will  tend  to  preserve  its  parallelism  w' 
with  the  plane  ACP ;  but  the  merid- 
ian of  the  place  will  have  the  posi- 
tion AP'C ;  that  is,  the  plane  of  vibra- 
tion will  now  make  an  angle  AP'B, 
or  PAP',  with  the  plane  of  the  meridian.  If  the  angle  PAP'  be 
taken  very  small,  the  figure  PAP'  may  be  regarded  as  a  plane 

triangle,  and  the  sine  of  the  angle  PAP',  or  the  angle  PAP',  will 

pp/ 
be  equal  to  -T-TJ.     But  PP',  which  is  the  distance  between  the  two 

meridians  measured  on  a  parallel  of  latitude,  varies  as  the  cosine 
of  the  latitude  (Loomis's  Trigonometry,  Art.  197,  Cor.) ;  and  AP 
is  the  tangent  of  PN,  or  the  cotangent  of  the  latitude.  Hence 

the  angle  PAP'  varies  as  — '—. — - ;  that  is,  as  the  sine  of  the  lati- 
cot.  lat ' 

tude ;  or  the  apparent  motion  of  the  plane  of  vibration  is  every 
where  proportional  to  the  sine  of  the  latitude  of  the  place. 

The  hourly  motion  of  the  plane  of  vibration  of  a  pendulum  set 
up  at  New  Haven  is  therefore  equal  to  15°  X  sin.  41°  18',  which  is 
a  little  less  than  10°  per  hour. 

If  we  suppose  tangents  to  be  drawn  to  each  meridian  circle  at 
its  intersection  with  the  parallel  PP',  all  these  tangents  will  inter- 
sect at  the  point  A.  These  tangents  lie  on  the  surface  of  a  cone 
whose  base  is  the  parallel  PP'.  If  we  suppose  the  surface  of  this 
cone  to  be  spread  out  upon  a  plane,  it  will  form  a  sector  of  a 
circle,  and  the  angle  at  the  centre  is  equal  to  the  motion  of  the 
plane  of  vibration  of  the  pendulum  in  24  hours  at  a  place  on  the 
parallel  PP'. 

It  is  indispensable  to  the  success  of  this  experiment  that  the 
pendulum  should  commence  its  vibration  without  any  lateral 
motion.  For  this  purpose,  the  pendulum  is  drawn  out  of  the 
vertical  position,  and  tied  to  a  fixed  object  by  a  fine  thread. 

C 


34  ASTRONOMY. 

9 

When  the  ball  is  quite  at  rest,  the  thread  is  burned,  and  the  pen- 
dulum commences  its  vibrations.  Experiments  of  this  kind  have 
been  made  at  numerous  places,  and  the  observed  rate  of  motion 
coincides  very  accurately  with  the  computed  rate,  and  this  coin- 
cidence may  be  regarded  as  a  direct  proof  that  the  earth  makes 
one  rotation  upon  its  axis  in  24  sidereal  hours. 

63.  Second  proof  of  the  earth's  rotation. — A  second  proof  of  the 
earth's  rotation  is  derived  from  the  motion  of  falling  bodies.     If 
.  the  earth  had  no  rotation  upon  an  axis,  a  heavy  body  let  fall  from 
any  elevation  would  descend  in  the  direction  of  a  vertical  line. 
But  if  the  earth  rotates  on  an  axis,  then,  since  the  top  of  a  tower 
describes  a  larger  circle  than  the  base,  its  easterly  motion  must  be 
more  rapid  than  that  of  the  base.     And  if  a  ball  be  dropped  from 
the  top  of  the  tower,  since  it  has  already  the  easterly  motion 
which  belongs  to  the  top  of  the  tower,  it  will  retain  this  easterly 
motion  during  its  descent,  and  its  deviation  to  the  east  of  the  ver- 
tical line  will  be  nearly  equal  to  the  excess  of  the  motion  of  the 
top  of  the  tower  above  that  of  the  base,  during  the  time  of  fall. 
Let  AB  represent  a  vertical  tower,  and  AA'  the  space  through 
rig.  14.      which  the  point  A  would  be  carried  by  the  earth's  ro- 
tation in  the  time  that  a  heavy  body  would  descend 
through  AB.     A  body  let  fall  from   the  top  of  the 
tower  will  retain  the  horizontal  velocity  which  it  had 
at  starting,  and,  when  it  reaches  the  earth's  surface^ 
will  have  moved  over  a  horizontal  space,  BD,  nearly 
equal  to  AA'.     But  the  foot  of  the  tower  will  have 
moved  only  through  BB',  so  that  the  body  will  be 
found  to  the  east  of  the  tower  by  a  space  equal  to  B'D 
nearly.     This  space  B'D,  for  an  elevation  of  500  feet, 
c       in  the  latitude  of  New  Haven,  is  a  little  over  one  inch, 
so  that  it  must  be  impossible  to  detect  this  deviation  except  from 
experiments  conducted  with  the  greatest  care  and  from  an  eleva- 
tion of  several  hundred  feet. 

54.  Results  of  experiments. — Numerous  experiments  have  been 
made  for  the  purpose  of  detecting  the  deviation  of  falling  bodies 
from  a  vertical  line.  In  the  year  1791  this  method  was  first  tried 
at  Bologna,  in  Italy,  from  a  tower  whose  height  was  256  English 
feet.  The  results  were  not  satisfactory,  the  experiments  not  hav- 
ing been  conducted  with  the  requisite  care. 


ARTIFICIAL  GLOBES.  85 

In  1802  the  experiments  were  repeated  at  Hamburg,  from  a 
tower  whose  height  was  250  English  feet,  and  the  observed  devia- 
tion from  the  vertical  differed  only  0.01  inch  from  the  deviation 
computed  by  theory.  In  1804  the  experiments  were  repeated  near 
Diisseldorf,  in  a  mine  whose  depth  was  280  feet,  and  the  observed 
deviation  differed  only  0.04  inch  from  the  computed  deviation. 

The  most  satisfactory  experiments  were  made  in  1832  at 
Freyberg,  Saxony,  in  a  mine  whose  depth  was  520  English  feet. 
According  to  the  mean  of  106  trials,  the  easterly  deviation  was 
1.12  inch,  while  the  deviation  by  theory  should  have  been  1.08 
inch.  The  experiments  also  showed  a  southerly  deviation  of  0.17 
inch,  which  is  not  accounted  for  by  theory. 

These  experiments  must  be  regarded  as  proving  that  the  earth 
does  rotate  upon  an  axis,  although  the  results  exhibit  discrepan- 
cies greater  than  might  have  been  anticipated,  and  which,  per- 
haps, are  not  fully  explained. 

ARTIFICIAL  GLOBES. 

55.  Artificial  globes  are  either  terrestrial  or  celestial.  The 
former  exhibits  a  miniature  representation  of  the  earth,  the  lat- 
ter exhibits  the  relative  position  of  the  fixed  stars.  The  mode 
of  mounting  is  usually  the  same  for  both,  and  many  of  the  cir- 
cles are  the  same  for  both  globes.  An  artincial  globe  is  mount- 
ed on  an  axis  which  is  supported  by  a  brass  ring,  which  repre- 
sents a  meridian,  and  is  called  the  brass  meridian.  This  ring  is 
supported  in  a  vertical  position  by  a  frame  in  such  a  manner 
that  the  axis  of  the  globe  can  be  inclined  at  any  angle  to  the 
horizon.  The  brass  meridian  is  graduated  into  degrees,  which 
are  numbered  from  the  equator  toward  either  pole.  The  horizon 
is  represented  by  a  broad  ring,  whose  plane  passes  through  the 
centre  of  the  globe.  It  is  also  graduated  into  degrees,  which  are 
numbered  in  both  directions  from  the  north  and  south  points,  to 
denote  azimuths;  and  there  is  usually  another  set  of  numbers 
which  begin  from  the  east  and  west  points,  to  denote  amplitudes. 
It  also  usually  contains  the  signs  of  the  ecliptic,  showing  the 
sun's  place  for  every  day  in  the  year. 

On  the  terrestrial  globe,  hour  circles  are  represented  by  great 
circles  drawn  through  the  poles  of  the  equator;  and  on  the  ce- 
lestial globe  corresponding  circles  are  drawn  through  the  poles 
of  the  ecliotiCj  and  a  series  of  small  circles  parallel  to  the  ecliptic 


36  ASTRONOMY. 

are  drawn  at  intervals  of  ten  degrees.  These  are  for  determining 
celestial  latitude  and  longitude.  The  ecliptic,  tropics,  and  polar 
circles  are  drawn  upon  the  terrestrial  globe,  as  well  as  upon  the 
celestial. 

About  the  north  pole  is  a  small  circle,  graduated  so  as  to  indi- 
cate hours  and  minutes,  while  a  small  index,  attached  to  the  brass 
meridian,  points  to  one  of  the  divisions  upon  this  hour  circle. 
This  index  can  be  moved  so  as  to  be  set  in  any  required  posi- 
tion. 

There  is  usually  a  flexible  strip  of  brass,  equal  in  length  to  one 
quarter  of  the  circumference  of  the  globe,  which  is  graduated  into 
degrees,  and  may  be  applied  to  the  surface  of  the  globe  so  as  to 
measure  the  distance  between  two  places,  or  the  altitude  of  any 
point  above  the  wooden  horizon.  Ilence  it  is  usually  called  the 
quadrant  of  altitude. 

PROBLEMS  ON  THE  TERRESTRIAL  GLOBE. 

56.  To  find  the  latitude  and  longitude  of  a  given  place. 

Turn  the  globe  so  as  to  bring  the  place  to  the  graduated  side 
of  the  brass  meridian ;  then  the  degree  of  the  meridian  directly 
over  the  place  will  indicate  the  latitude,  and  the  degree  on  the 
equator  under  the  brass  meridian  will  indicate  the  longitude. 

Example.  What  are  the  latitude  and  longitude  of  Cape  Horn  ? 

57.  Given  the  latitude  and  longitude,  to  find  the  place. 

Bring  the  degree  of  longitude  on  the  equator  under  the  brass 
meridian,  then  under  the  given  latitude  on  the  brass  meridian 
will  be  found  the  place  required. 

Example.  Find  the  place  which  is  situated  in  Lat.  30°  N.  and 
Long.  90°  W. 

58.  To  find  the  bearing  and  distance  from  one  place  to  anotfier  on 
the  earth's  surface. 

Elevate  the  north  pole  to  the  latitude  of  the  first -mentioned 
place,  and  bring  this  place  to  the  brass  meridian.  Screw  the 
quadrant  of  altitude  to  this  point  of  the  brass  meridian,  and  make 
it  pass  through  the  other  place.  Then  the  bearing  of  the  second 
place  from  the  first  will  be  indicated  on  the  wooden  horizon,  and 
the  number  of  degrees  on  the  quadrant  of  altitude  will  show  the 
distance  between  the  two  places  in  degrees,  which  may  be  reduced 


ARTIFICIAL  GLOBES.  37 

to  miles  by  multiplying  them  by  69-j,  because  69^  miles  make 
nearly  one  degree. 

Example.  What  is  the  bearing  and  distance  of  Liverpool  from 
New  York? 

59.  To  find  the  antipodes  of  a  given,  place. 

Bring  the  given  place  to  the  wooden  horizon,  and  the  opposite 
point  of  the  horizon  will  indicate  the  antipodes.  The  one  place 
will  be  as  far  from  the  north  point  of  the  wooden  horizon,  as  the 
other  is  from  the  south  point. 

Example.  Find  the  antipodes  of  London. 

60.  Given  the  hour  of  the  day  at  any  place,  to  find  the  liour  at  any 
other  place. 

Bring  the  first-mentioned  place  to  the  brass  meridian,  and  set 
the  hour  index  to  the  given  time.  Turn  the  globe  till  the  other 
place  comes  to  the  meridian ;  the  hour  circle  will  show  the  re- 
quired time. 

Example.  What  time  is  it  at  San  Francisco  when  it  is  10  A.M. 
in  New  York  ? 

61.  To  find  the  time  of  the  surCs  rising  and  setting  at  a  given  place, 
on  a  given  day. 

Elevate  the  pole  to  the  latitude  of  the  place.  On  the  wooden 
horizon  find  the  day  of  the  month,  and  against  it  is  given  the 
sun's  place  in  the  ecliptic,  expressed  in  signs  and  degrees.  Bring 
the  sun's  place  to  the  meridian,  and  set  the  hour  index  to  12. 
Turn  the  globe  till  the  sun's  place  is  brought  down  to  the  east- 
ern horizon ;  the  hour  index  will  show  the  time  of  rising.  Turn 
the  globe  till  the  sun's  place  comes  to  the  western  horizon ;  the 
hour  index  will  tell  the  time  of  setting. 

Example.  Eequired  the  time  of  rising  and  setting  of  the  sun  at 
Washington,  August  18th. 


22S803 


38  ASTRONOMY. 


CHAPTER  II. 

INSTRUMENTS  FOR  OBSERVATION. — THE  CLOCK. — TRANSIT  IN- 
STRUMENT. —  MURAL  CIRCLE.  —  ALTITUDE  AND  AZIMUTH  IN- 
STRUMENT, AND  THE  SEXTANT. 

62.  Why  observations  are  chiefly  made  in  the  meridian. — When- 
ever'circumstances  allow  an  astronomer  to  select  bis  own  time  of 
observation,  almost  all  his  observations  of  the  heavenly  bodies  are 
made  when  they  are  upon  the  meridian,  because  a  large  instru- 
ment can  be  more  accurately  and  permanently  adjusted  to  de- 
scribe a  vertical  plane  than  any  plane  oblique  to  the  horizon ;  and 
there  is  no  other  vertical  plane  which  combines  so  many  advant- 
ages as  the  meridian.    The  places  of  the  heavenly  bodies  are  most 
conveniently  expressed  by  right  ascension  and  declination,  and 
the  right  ascension  is  simply  the  time  of  passing  the  meridian,  as 
shown  by  a  sidereal  clock.     Moreover,  when  a  heavenly  body  is 
at  its  upper  culmination,  its  refraction  and  parallax  are  the  least 
possible;  and  in  this  position  refraction  and  parallax  do  not  af- 
fect the  right  ascension  of  the  body,  but  simply  its  declination; 
while  for  every  position  out  of  the  meridian,  they  affect  both 
right  ascension  and  declination. 

63.  The  Clock. — The  standard  instruments  of  an  astronomical 
observatory  are  the  clock,  the  transit  instrument,  and  the  mural 
circle. 

In  a  stationary  observatory,  a  pendulum  clock  is  used  for  meas- 
uring time.  The  clock  should  be  so  regulated  that  if  a  star  be 
observed  upon  the  meridian  at  the  instant  when  the  hands  point 
to  Oh.  Om.  Os.,  they  "will  point  to  Oh.  Om.  Os.  when  the  same  star  is 
next  seen  on  the  meridian.  This  interval  is  called  a  sidereal  day, 
and  is  divided  into  24  sidereal  hours.  If  the  clock  were  perfect, 
the  pendulum  would  make  86,400  vibrations  in  the  interval  be- 
tween two  successive  returns  of  the  same  star  to  the  meridian. 
But  no  clock  is  perfect,  and  it  is  therefore  necessary  to  determine 
the  error  and  rate  of  the  clock  daily,  and  in  all  our  observations 
to  make  an  allowance  for  the  error  of  the  clock. 


THE  TRANSIT  INSTRUMENT. 


39 


The  error  of  a  clock  at  any  time  is  its  difference  from  true  side- 
real time.  The  rate  of  the  clock  is  the  change  of  its  error  in  24 
hours.  Thus,  if,  on  the  8th  of  January,  when  Aldebaran  passed 
the  meridian,  the  clock  was  found  to  be  30.84s.  slow,  and  on  the 
9th  of  January,  when  the  same  star  passed  the  meridian,  the  clock 
was  31.66s.  slow,  the  clock  lost  0.82s.  per  day.  In  other  words,  the 
error  of  the  clock  January  9th  was  —31.66s,  and  its  daily  rate 
-0.82s. 

The  Transit  Instrument. 

64.  Most  of  the  observations  of  the  heavenly  bodies  are  made 
when  they  are  upon  the  celestial  meridian ;  and,  in  many  cases, 
the  sole  business  of  the  observer  is  to  determine  the  exact  instant 
when  the  object  is  brought  to  the  meridian,  by  the  apparent  di- 
urnal motion  of  the  firmament.  This  phenomenon  of  passing  the 
meridian  is  called  a  transit,  and  an  instrument,  mounted  in  such  a 
manner  as  to  enable  an  observer,  supplied  with  a  clock,  to  ascer- 
tain the  exact  time  of  transit,  is  called  a  transit  instrument. 


65.  Description  of  the  Tran- 
sit Instrument. — Such  an  in- 
strument consists  of  a  tele- 
scope, TT,  mounted  upon  an 
axis,  AB,  at  right  angles  to 
the  tube,  which  axis  occupies 
a  horizontal  position,  and 
points  east  and  west.  The 
tube  of  the  telescope,  when 
horizontal,  will  therefore  be 
directed  north  and  south ;  and 
if  the  telescope  be  revolved 
on  its  axis  through  180°,  the 
central  line  of  the  tube  will 
move  in  the  plane  of  the  me- 
ridian, and  may  be  directed 
to  any  point  on  the  celestial 
meridian. 

For  a  large  transit  instru- 
ment, two  stone  piers,  PP,  are 
erected  on  a-  solid  foundation, 


Fig  15 


40  ASTRONOMY. 

standing  on  an  east  and  west  line.  On  the  top  of  each  of  the 
piers  is  secured  a  metallic  support,  in  the  form  of  the  letter  Y,  to 
receive  the  extremities  of  the  axis  of  the  telescope.  At  the  left 
end  of  the  axis  there  is  a  screw,  by  which  the  Y  of  that  extrem- 
ity may  be  raised  or  lowered  a  little,  in  order  that  the  axis  may 
be  made  perfectly  horizontal.  At  the  right  end  of  the  axis  is  a 
screw,  by  which  the  Y  of  that  extremity  may  be  moved  backward 
or  forward,  in  order  to  enable  us  to  bring  the  telescope  into  the 
plane  of  the  meridian.  In  order  that  the  pivots  of  the  axis  may 
be  relieved  from  a  portion  of  the  weight  of  the  instrument,  there 
is  raised  upon  the  top  of  each  pier  a  brass  pillar  supporting  a 
lever,  from  one  end  of  which  hangs  a  hook  passing  under  one 
extremity  of  the  axis,  while  a  counterpoise  sliding  on  the  other 
end  of  the  lever  may  be  made  to  support  as  much  of  the  weight 
of  the  instrument  as  is  desired. 

66.  The  Spirit  Level. — When  the  instrument  is  properly  adjust- 
ed, its  axis  will  be  horizontal,  and  directed  due  east  and  west.    If 
the  axis  be  not  exactly  horizontal,  its  deviation  may  be  ascer- 
tained by  placing  upon  it  a  spirit  level.     This  consists  of  a  glass 

Fi<T  1C  tube,  AB,  nearly  filled  with  alcohol 

c  D  __D___  or  ether.  The  tube  forms  a  portion 

j/T^= ^^~"H~?lii  °f  a  rmS  °f  a  verJ  large  radius,  and 

1 -==v  when  it  is  placed  horizontally,  with 

its  convexity  upward,  the  bubble,  CD,  will  occupy  the  highest  po- 
sition in  the  middle  of  its  length.  A  graduated  scale  is  attached 
to  the  tube,  by  which  we  may  measure  any  deviation  of  the  bub- 
ble from  the  middle  of  the  tube. 

To  ascertain  whether  the  axis  of  the  telescope  is  horizontal,  ap- 
ply the  level  to  it,  and  see  if  the  bubble  occupies  the  middle  of 
the  tube.  If  it  does  not,  one  end  must  be  elevated  or  depressed. 
In  order  to  accomplish  this,  one  of  the  supports  of  the  axis  is  con- 
structed so  as  to  be  moved  vertically  through  a  small  space  by 
means  of  a  fine  screw.  The  level  must  now  be  taken  up  and  re- 
versed end  for  end,  and  this  operation  must  be  repeated  until  the 
bubble  rests  in  the  middle  of  the  tube  in  both  positions  of  the 
level. 

67.  Mttiiod  of  observing  trajtsits. — In  the  focus  of  the  eye-piece 
of  the  transit  instrument,  at  F,  is  placed  a  system  of  5  or  7  equi- 


METHOD   OF   OBSERVING   TRANSITS. 


Fig.  IT. 

X 


X 


distant  and  vertical  wires,  intersected  by  1  or 
2  horizontal  wires.  When  the  instrument  has 
been  properly  adjusted,  the  middle  wire,  MN, 
will  be  in  the  plane  of  the  meridian,  and  when 
an  object  is  seen  upon  it,  this  object  will  be 
on  the  celestial  meridian.  The  fixed  stars  ap- 
pear in  the  telescope  as  bright  points  of  light 
without  sensible  magnitude,  and  by  the  di- 
urnal motion  of  the  heavens  a  star  is  carried  successively  over 
each  of  the  wires  of  the  transit  instrument.  The  observer,  just 
before  the  star  enters  the  field  of  view,  writes  down  the  hour  and 
minute  indicated  by  the  clock,  and  proceeds  to  count  the  seconds 
by  listening  to  the  beats  of  the  clock,  while  his  eye  is  looking 
through  the  telescope.  He  observes  the  instant  at  which  the  star 
crosses  each  of  the  wires,  estimating  the  time  to  the  nearest  tenth 
of  a  second ;  and  by  taking  a  mean  of  all  these  observations,  he 
obtains  with  great  precision  the  instant  at  which  the  star  passed 
the  middle  wire,  and  this  is  regarded  as  the  true  time  of  the 
transit.  The  mean  of  the  observations  over  several  wires,  is  con- 
sidered more  reliable  than  an  observation  over  a  single  wire. 

In  many  observatories  it  is  now  customary  to  employ  the  elec- 
tric circuit  to  record  transit  observations.  By  pressing  the  finger 
upon  a  key  at  the  instant  a  star  is  seen  to  pass  one  of  the  wires 
of  the  transit,  a  mark  is  made  upon  a  sheet  of  paper  which  is 
graduated  into  seconds  by  the  pendulum  of  the  observatory  clock, 
according  to  the  mode  more  fully  explained  in  Art.  337. 

During  the  day,  the  wires  are  visible  as  fine  black  lines  stretch- 
ed across  the  field  of  view.  At  night  they  are  rendered  visible 
by  a  lamp,  L,  by  which  the  field  of  view  is  faintly  illumined. 

"When  we  observe  the  sun  or  any  object  which  has  a  sensible 
disc,  the  time  of  transit  is  the  instant  at  which  the  centre  of  the 
disc  crosses  the  middle  wire.  This  time  is  obtained  by  observing 
the  instants  at  which  the  eastern  and  western  edges  of  the  disc 
touch  each  of  the  wires  in  succession,  and  taking  the  mean  of  all 
the  observations.  When  the  visible  disc  is  not  circular,  special 
methods  of  reduction  are  employed. 

• 

68.  Rate  of  the  diurnal  motion. — Since  the  celestial  sphere  re« 
voives  at  the  rate  of  15°  per  hour,  or  15  seconds  of  arc  in  one 
second  of  time,  the  space  passed  over  between  two  successive 


42  ASTRONOMY, 

beats  of  the  pendulum  will  be  15"  of  arc.  When  the  sun  is  on 
§he  equator,  and  its  apparent  diameter  is  32'  of  arc,  the  interval 
between  the  contacts  of  the  east  and  west  limbs  with  the  middle 
wire  will  be  2m.  8s. 

69.  To  adjust  a  transit  instrument  to  the  meridian. — A  transit  in- 
strument may  be  adjusted  to  describe  the  plane  of  the  meridian, 
by  observations  of  the  pole  star.    Direct  the  telescope  to  the  pole 
star  at  the  instant  of  its  crossing  the  meridian,  as  near  as  the  time 
can  be  ascertained.     The  transit  will  then  be  nearly  in  the  plane 
of  the  meridian.     Having  leveled  the  axis,  turn  the  telescope  to 
a  star  about  to  cross  the  meridian,  near  the  zenith.     Since  every 
vertical  circle  intersects  the  meridian  at  the  zenith,  a  zenith  star 
will  cross  the  field  of  the  telescope  at  the  same  time,  whether  the 
plane  of  the  transit  coincide  with  the  meridian  or  not.     At  the 
moment  the  star  crosses  the  central  wire,  set  the  clock  to  the  star's 
right  ascension  which  is  given  by  the  star  catalogues,  and  the 
clock  will  henceforth  indicate  nearly  sidereal  time.    The  approxi- 
mate times  of  the  upper  and  lower  culminations  of  the  pole  star 
are  then  known.    Observe  the  pole  star  at  one  of  its  culminations, 
following  its  motion  until  the  clock  indicates  its  right  ascension, 
or  its  right  ascension  plus  12  hours.     Move  the  whole  frame  of 
the  transit  so  that  the  central  wire  shall  coincide  nearly  with  the 
star,  and  complete  the  adjustment  by  means  of  the  azimuth  screw. 
The  central  wire  will  now  coincide  almost  precisely  with  the  me- 
ridian of  the  place. 

70.  Final  verification. — The  axis  being  supposed  perfectly  hori- 
zontal, if  the  middle  wire  of  the  telescope  is  exactly  in  the  merid- 
ian, it  will  bisect  the  circle  which  the  pole  star  describes  in  24  si- 
dereal hours  round  the  polar  point.     If,  then,  the  interval  between 
the  upper  and  lower  culminations  is  exactly  equal  to  the  interval 
between  the  lower  and  upper,  the  adjustment  is  complete.     But 
if  the  time  elapsed  while  the  star  is  traversing  the  eastern  semi- 
circle, is  greater  than  that  of  traversing  the  western,  the  plane  in 
which  the  telescope  moves  is  westward  of  the  true  meridian  on 
the  north  horizon  ;  and  vice  versa  if  the  western  interval  is  great- 
est.   This  error  of  position  must  be  corrected  by  turning  the  azi- 
muth screw.     The  adjustment  must  then  be  verified  by  further 
observations,  until,  by  continued  approximations,  the  instrument 
is  fixed  correctly  in  the  meridian. 


THE   MURAL   CIRCLE.  43 

Other  methods  of  adjusting  a  transit  instrument  to  the  plane 
of  the  meridian,  will  be  found  in  works  specially  devoted  to  Prac- 
tical Astronomy. 

The  Mural  Circle. 

71.  The  mural  circle  is  a  graduated  circle,  aaaa,  usually  made 
of  brass,  and  having  an  axis  passing  through  its  centre.  This  axis 
should  be  exactly  horizontal ;  and  it  is  supported  by  a  stone  pier 
or  wall,  so  as  to  be  directed  due  east  and  west.  To  the  circle  is 
attached  a  telescope,  MM,  so  that  the  entire  instrument,  including 
the  telescope,  turns  in  the  plane  of  the  meridian. 

Fig.  18. 


Mural  circles  have  been  made  eight  feet  in  diameter,  but  gen- 
erally they  have  been  made  six  feet ;  and  at  present  astronomers 
are  pretty  well  agreed  that  a  circle  of  five  feet  is  better  than  any 
larger  size,  being  less  liable  to  change  of  form  from  its  great 
weight.  At  the  great  Russian  observatory  at  Pulkova,  the  largest 
circle  employed  is  only  four  feet  in  diameter.  The  circle  is  di- 
vided into  degrees,  and  subdivided  into  spaces  of  five  minutes, 
and  sometimes  of  two  minutes,  the  divisions  being  numbered  from 
0°  to  360°  round  the  entire  circle.  The  smallest  spaces  on  the 


44  ASTRONOMY. 

limb  are  further  subdivided  to  single  seconds,  sometimes  by  a  ver- 
nier, but  generally  by  a  reading  microscope. 

72.  Use  of  the  Vernier.— A  vernier  is  a  scale  of  small  extent, 
graduated  in  such  a  manner  that,  being  moved  by  the  side  of  a 
fixed  scale,  we  are  enabled  to  measure  minute  portions  of  this 
scale.  The  length  of  this  movable  scale  is  equal  to  a  certain  num- 
ber of  parts  of  that  to  be  subdivided ;  but  it  is  divided  into  parts 
either  one  more,  or  one  less,  than  those  of  the  primary  scale  taken 
for  the  length  of  the  vernier.  Thus,  if  we  wish  to  measure  hund- 
redths  of  an  inch,  as  in  the  case  of  a  barometer,  we  first  divide  an 
inch  into  ten  equal  parts.  We  then  construct  a  vernier  equal  in 
length  to  11  of  these  divisions,  but  divide  it  into  10  equal  parts, 
by  which  means  each  division  on  the  vernier  is  -^th  longer  than 
a  division  of  the  primary  scale. 

rig.  19.  Thus,  let  AB  be  the  upper  end  of  a  ba 

rometer  tube,  the  mercury  standing  at  the 
point  C ;  the  scale  is  divided  into  inches  and 
-50  tenths  of  an  inch,  and  the  middle  piece,  num- 
bered from  1  to  9,  is  the  vernier,  that  may 
be  slid  up  or  down,  and  having  10  of  its  di- 
visions equal  to  11  divisions  of  the  scale; 
that  is,  to  -|4ths  of  an  inch.  Therefore,  each 
division  of  the  vernier  is  -^^ths  of  an  inch ; 
or  one  division  of  the  vernier  exceeds  one 
division  of  the  scale,  by  Trc-th  of  an  inch. 
Now,  as  the  sixth  division  of  the  vernier  (in 
the  figure)  coincides  with  a  division  of  the 
scale,  the  fifth  division  of  the  vernier  will 
of  an  inch  above  the  nearest  division  of  the  scale ; 
the  fourth  division  -rfo-ths  of  an  inch ;  and  the  top  of  the  vernier 
will  be  -nnrths  of  an  inch  above  the  next  lower  division  of  the 
scale ;  ?'.  e.,  the  top  of  the  vernier  coincides  with  29.66  inches  upon 
the  scale.  In  practice,  therefore,  we  observe  what  division  of  the 
vernier  coincides  with  a  division  of  the  scale ;  this  will  show  the 
hundredths  of  an  inch  to  be  added  to  the  tenths  next  below  the 
vernier  at  the  top. 

73.  Vernier  applied  to  graduated  circles.     The  limb  of  a  sextant 
is  usually  divided  into  parts  of  10'  each ;  and  if  a  vernier  be  taken 


stand 


THE   READING   MICROSCOPE.  45 

equal  in  length  to  59  of  these  parts,  and  be  subdivided  into  60  equal 
portions,  each  division  of  the  vernier  will  be  equal  to  %%  of  10', 
while  each  division  of  the  limb  is  equal  to  10'.  Hence  an  interval 
on  the  limb  exceeds  an  interval  on  the  vernier  by  7V  of  10';  that 
is,  by  10".  Hence,  although  the  limb  is  graduated  into  parts  equal 
to  10',  the  vernier  enables  us  to  measure  angles  as  small  as  10". 
A  circle  of  two  or  three  feet  radius  may  be  graduated  into 
spaces  equal  to  2'  each,  and  by  means  of  a  vernier  we  may  measure 
angles  as  small  as  2"  or  even  I".  When  angles  as  small  as  V 
are  to  be  measured,  it  is  difficult  to  tell  which  division  of  the 
vernier  coincides  with  a  division  of  the  limb,  and  the  vernier  is 
less  convenient  than  the  reading  microscope. 

74.  The  reading  Microscope. — The  large  circles  employed  in  as- 
tronomical observations  are  divided  into  spaces  as  small  as  5',  and 
sometimes  as  small  as  2'.  By  a  vernier  these  spaces  are  some- 
times subdivided  so  as  to  give  single  seconds.  The  vernier  is 
generally  employed  in  instruments  made  by  German  artists,  but 
upon  large  circles  made  by  English  artists  the  subdivisions  are 
usually  effected  by  the  reading  microscope.  Fig.  20  represents 
the  appearance  of  one  of  A  Fig.2o. 

these  microscopes.  It  is 
a  compound  microscope, 
consisting  of  three  lenses, 
one  of  which  is  the  ob- 
ject lens  at  L,  and  the 
other  two  are  formed  into 
a  positive  eye-piece,  GH. 
In  the  common  focus  of  the  object  lens  and  the  eye-piece  at  K,  is 
placed  the  spider-line  micrometer.  It  consists  of  a  small  rectan- 
gular frame,  across  which  are  stretched  two  spider-lines  forming 
an  acute  cross,  and  is  moved  laterally  by  means  of  a  screw,  M. 
The  figure  on  the  right  shows  the  field  of  view,  with  the  magni- 
fied divisions  on  the  instrument,  as  seen  through  the  microscope. 
When  the  microscope  is  properly  adjusted,  the  image  of  the  di- 
vided limb  and  the  spider-lines  are  distinctly  visible  together; 
and  also  five  revolutions  of  the  screw  must  exactly  measure  one 
of  the  5'  spaces  on  the  limb.  One  revolution  of  the  head  of  the 
screw  will  therefore  carry  the  spider-lines  over  a  space  of  1'.  The 
circumference  of  the  circle  attached  to  the  head,  M,  is  divided  into 


46  ASTRONOMY. 

60  equal  parts,  so  that  the  motion  of  the  head  through  one  of  these 
divisions,  advances  the  spider-lines  through  a  space  of  I".  There 
are  six  of  these  microscopes,  A,  B,  C,  D,  E,  F,  placed  at  equal  dis- 
tances round  the  circle,  and  firmly  attached  to  the  pier. 

75.  To  determine  the  horizontal  point. — In  order  to  ascertain  the 
horizontal  point  upon  the  limb  of  the  circle,  we  direct  the  telescope 
upon  any  star  which  is  about  crossing  the  meridian,  and  bring  its 
image  to  coincide  with  the  horizontal  wire  which  passes  through 
the  centre  of  the  field  of  the  telescope.     The  graduation  is  then 
read  off  by  the  fixed  microscopes.    On  the  next  night,  we  place  a 
vessel  containing  mercury  in  a  convenient  position  near  the  floor, 
so  that,  by  directing  the  telescope  of  the  mural  circle  toward  it, 
the  same  star  may  be  seen  reflected  from  the  surface  of  the  mer- 
cury, and  we  bring  the  reflected  image  to  coincide  with  the  hori- 
zontal wire  of  the  telescope.     The  graduation  is  then  read  off  as 
before.    Now,  by  a  law  of  optics,  the  reflected  image  will  appear 
as  much  below  the  horizon  as  the  star  is  really  above  the  horizon ; 
therefore  half  the  sum  of  the  two  readings  at  either  of  the  micro- 
scopes, will  be  the  reading  at  the  same  microscope  when  the  tel- 
escope is  horizontal. 

76.  To  determine  the  altitude  of  any  object. — Having  determined 
the  reading  of  each  of  the  microscopes  when  the  telescope  is  di- 
rected to  the  horizon,  if  we  wish  to  determine  the  altitude  of  any 
object,  we  direct  the  telescope  to  it,  so  that  it  may  be  seen  on  the 
horizontal  wire  as  the  star  passes  the  meridian,  and  then  read  off 
the  microscopes.    The  difference  between  the  last  reading,  and  the 
reading  when  the  telescope  is  horizontal,  is  the  altitude  required. 

The  zenith  distance  of  an  object  is  found  by  subtracting  its  alti- 
tude from  90°. 

The  pole  star  crosses  the  meridian,  above  and  below  the  pole, 
at  intervals  of  12  hours  sidereal  time;  apd  the  true  position  of  the 
pole  is  exactly  midway  between  the  two  points  where  the  star 
crossi-s  the  meridian;  therefore  half  the  sum  of  the  readings  of 
either  microscope  when  the  pole  star  makes  its  transit'  above  and 
below  the  pole,  will  be  the  reading  for  the  pole  itself. 

The  readings  for  the  pole  being  determined,  those  which  cor- 
respond to  the  point  where  the  celestial  equator  crosses  the  merid- 
ian, are  easily  found,  since  the  equator  is  90°  from  the  pole. 


ALTITUDE  AND  AZIMUTH  INSTRUMENT. 


47 


Having  determined  the  position  of  the  celestial  equator,  the  dec- 
lination of  any  star  is  easily  determined,  since  its  declination  is 
simply  its  distance  from  the  equator. 

77.  The  Transit  Circle, — Since  the  mural  circle  has  a  short  axis, 
its  position  in  the  meridian  is  unstable,  and  therefore  it  can  not 
be  relied  upon  to  give  the  right  ascension  of  stars  with  great  ac- 
curacy.    It  was  formerly  thought  necessary  at  Greenwich  to  have 
two  instruments  for  determining  a  star's  place ;  viz.,  a  transit  in- 
strument to  determine  its  right  ascension,  and  a  mural  circle  to 
determine  its  declination.     The  German  astronomers  have,  how- 
ever, combined  both  instruments  in  one,  under  the  name  of  me- 
ridian circle,  which  is  essentially  the  transit  instrument  already 
described,  with  a  large  graduated  circle  attached  to  its  axis;  and 
a  large  transit  circle  is  rig.  21. 

now  in  use  at  the  Green- 
wich Observatory. 

Altitude  and  Azimuth  In- 
strument. 

78.  The  altitude  and 
azimuth  instrument  con- 
sists  of  one    graduated 
circle  confined  to  a  hori- 
zontal plane;    a  second 
graduated  circle  perpen- 
dicular  to    the   former, 
and    capable    of  being 
turned  into  any  azimuth ; 
and  a  telescope   firmly 
fastened  to  the   second 
circle,  and  turning  with 
it  in  altitude.     The  ap- 
pearance of  this  instru- 
ment   will    be    learned 
from  the  annexed  figure. 

EE  are  two  legs  of  the 
tripod  upon  which  the 
instrument  rests;  and  in 
close  contact  with  the 


48  ASTRONOMY. 

tripod  is  placed  the  azimuth  circle,  FF.  Above  the  azimuth  cir- 
cle,  and  concentric  with  it,  is  placed  a  strong  circular  plate,  which 
sustains  the  whole  of  the  upper  part  of  the  instrument,  and  also  a 
pointer,  to  show  the  degree  and  nearest  five  minutes  to  be  read 
off  on  the  azimuth  circle ;  the  remaining  minutes  and  seconds  be- 
ing obtained  by  means  of  the  two  reading  microscopes  C  and  D. 
The  pillars,  HIT,  support  the  transit  axis  I  by  means  of  the  pro- 
jecting pieces  LL.  The  telescope,  MM,  is  connected  with  the  hor- 
izontal axis  in  a  manner  similar  to  that  of  the  transit  instrument. 
Upon  the  axis,  as  a  centre,  is  fixed  the  double  circle  NN,  each 
circle  being  placed  close  against  the  telescope.  The  circles  are 
fastened  together  by  small  brass  pillars,  and  the  graduation  is 
made  on  a  narrow  ring  of  silver,  inlaid  on  one  of  the  sides,  which 
is  usually  termed  thence  of  the  instrument.  The  reading  micro- 
scopes, AB,  for  the  vertical  circle,  are  carried  by  two  arms,  PP, 
attached  near  the  top  of  one  of  the  pillars. 

In  the  principal  focus  of  the  telescope,  are  stretched  spider  lines, 
as  in  the  transit  instrument,  and  the  illumination  is  effected  in  a 
similar  manner. 

79.  Adjustments  of  the  instrument. — Before  commencing  obser- 
vations with  this  instrument,  the  horizontal  circle  must  be  leveled, 
and  also  the  axis  of  the  telescope.  The  meridional  point  on  the 
azimuth  circle  is  its  reading  when  the  telescope  is  pointed  north 
or  south,  and  may  be  determined  by  observing  a  star  at  equal  alti- 
tudes east  and  west  of  the  meridian,  and  finding  the  point  mid- 
way between  the  two  observed  azimuths ;  or  the  instrument  may 
be  adjusted  to  the  meridian  in  the  same  manner  as  a  transit.  The 
horizontal  point  of  the  altitude  circle  is  its  reading  when  the  axis 
of  the  telescope  is  horizontal,  and  may  be  found,  as  with  the  mu- 
ral circle,  by  alternate  observations  of  a  star  directly  and  reflected 
from  the  surface  of  mercury. 

This  instrument  lias  the  advantage  over  the  transit  instrument 
and  mural  circle,  in  its  being  able  to  determine  the  place  of  a  star 
in  any  part  of  the  visible  heavens;  but  we  ordinarily  require  the 
place  of  a  star  to  be  given  in  right  ascension  and  declination  in- 
stead of  altitude  and  azimuth,  and  to  deduce  the  one  from  the 
other  requires  a  laborious  computation.  Hence  the  altitude  and 
azimuth  instrument  is  but  little  used  in  astronomical  observations, 
except  for  special  purposes,  as,  for  example,  to  investigate  the  laws 
of  refraction. 


103 

THE   SEXTANT.  49 


The  Sextant. 

80.  The  arc  of  a  sextant,  as  its  name  implies,  contains  sixty  de- 
grees, but,  on  account  of  the  double  reflection,  is  divided  into  120 
degrees.  The  annexed  figure  represents  a  sextant,  the  frame  being 
generally  made  of  brass;  Fig. 22. 

the  handle,  H,  at  its  back,  is 
made  of  wood.  When  ob- 
serving, the  instrument  is 
to  be  held  with  one  hand 
by  the  handle,  while  the 
other  hand  moves  the  in- 
dex G.  The  arc,  AB,  is  di- 
vided into  120  or  more  de- 
grees, numbered  from  A  to- 
ward B,  and  each  degree  is 
divided  into  six  equal  parts 
of  10'  each,  while  the  ver- 
nier shows  10".  The  divisions  are  also  continued  a  short  dis- 
tance on  the  other  side  of  zero  toward  A,  forming  what  is  called 
the  arc  of  excess.  The  microscope,  M,  is  movable  about  a  centre, 
and  may  be  adjusted  to  read  off  the  divisions  on  the  graduated 
limb.  A  tangent  screw,  D,  is  fixed  to  the  index,  for  the  purpose 
of  making  the  contacts  more  accurately  than  can  be  done  by  hand. 
When  the  index  is  to  be  moved  a  considerable  distance,  the  screw 
I  must  be  loosened ;  and  when  the  index  is  brought  nearly  to  the 
required  division,  the  screw  I  must  be  tightened,  and  the  index 
be  moved  gradually  by  the  tangent  screw.  The  upper  end  of  the 
index  Gr  terminates  in  a  circle,  across  which  is  fixed  the  silvered 
index  glass  C,  over  the  centre  of  motion,  and  perpendicular  to  the 
plane  of  the  instrument.  To  the  frame  at  N  is  attached  a  second 
glass,  called  the  horizon  glass,  the  lower  half  of  which  only  is  sil- 
vered. This  must  also  be  perpendicular  to  the  plane  of  the  in- 
strument, and  in  such  a  position  that  its  plane  shall  be  parallel  to 
the  plane  of  the  index  glass  C,  when  the  vernier  is  set  to  zero  on 
the  limb  AB. 

The  telescope,  T,  is  carried  by  a  ring,  K ;  and  in  the  focus  of  the 
object  glass  are  placed  two  wires  parallel  to  each  other,  and  equi- 
distant from  the  axis  of  the  telescope.  Four  dark  glasses,  of  dif- 
ferent depths  of  shade  and  color,  are  placed  at  F,  between  the  in; 

D 


-V 


50  ASTRONOMY. 

dex  and  horizon  glasses ;  also  three  more  at  E,  any  one  or  more 
of  which  can  be  turned  down,  to  moderate  the  intensity  of  the 
light  before  reaching  the  eye,  when  a  bright  object,  as  the  sun,  is 
observed. 

SI.  To  measure  the  altitude  of  the  sun  by  reflection  from  mercury. 
—Set  the  index  near  zero.  Hold  the  instrument  with  the  right 
hand  in  the  vertical  plane  of  the  sun,  with  the  telescope  pointed 
toward  the  sun.  Two  images  will  be  seen  in  the  field  of  view, 
one  of  which,  viz.,  that  formed  by  reflection,  will  apparently  move 
downward  when  the  index  is  pushed  forward.  Follow  the  re- 
flected image  as  it  travels  downward,  until  it  appears  to  be  as  far 
below  the  horizon  as  it  was  at  first  above,  and  the  image  of  the 
sun  reflected  from  the  mercury  also  appears  in  the  field  of  view. 
Fasten  the  index,  and,  by  means  of  the  tangent  screw,  bring  the 
upper  or  lower  limb  of  the  sun's  image  reflected  from  the  index 
glass,  into  contact  with  the  opposite  limb  of  the  image  reflected 
from  the  artificial  horizon.  The  angle  shown  on  the  instrument, 
when  corrected  for  the  index  error,  will  be  double  the  altitude  of 
the  sun's  limb  above  the  horizontal  plane ;  to  the  half  of  which, 
if  the  semi-diameter,  refraction  and  parallax  be  applied,  the  result 
will  be  the  true  altitude  of  the  centre. 

If  the  observer  is  at  sea,  the  natural  horizon  must  be  employed. 
Direct  the  sight  to  that  part  of  the  horizon  beneath  the  sun,  and 
move  the  index  till  you  bring  the  image  of  its  lower  limb  to  touch 
the  horizon  directly  underneath  it. 

82.  To  measure  the  distance  between  two  objects. — To  find  the  dis- 
tance between  the  moon  and  sun,  hold  the  sextant  so  that  its  plane 
may  pass  through  both  objects.     Look  directly  at  the  moon 
through  the  telescope,  and  move  the  index  forward  till  the  sun's 
image  is  brought  nearly  into  contact  with  the  moon's  nearest  limb. 
Fix  the  index  by  the  screw  under  the  sextant,  and  make  the  con- 
tact perfect  by  means  of  the  tangent  screw.     The  index  will  then 
show  the  distance  of  the  nearest  limbs  of  the  sun  and  moon.     In 
a  similar  manner  may  we  measure  the  distance  between  the  moon 
and  a  star. 

83.  Dip  of  the  horizon. — In  observing  an  altitude  at  sea  with 
the  sextant,  the  image  of  an  object  is  made  to  coincide  with  the 


ATMOSPHERIC  REFRACTION".  51 

visible  horizon ;  but  since  the  eye  is  elevated  above  the  surface 
of  the  sea,  the  visible  horizon  will  be  below  the  true  horizontal 
plane. 

Let  AC  be  the  radius  of  the  earth,  AD 
the  height  of  the  eye  above  the  level  of  the 
sea,  ED  El  a  horizontal  plane  passing  through 
the  place  of  the  observer ;  then  IIDB  will  be 
the  dip  or  depression  of  the  horizon,  which 
may  be  found  as  follows : 

The  angle  HDB  is' equal  to  the  angle  BCD; 
and  in  the  right-angled  triangle  BCD,  BD2  = 
CD2-BC2=(AC  +  AD)2-AC2.  Whence  BD 
becomes  known.  Then,  in  the  same  triangle, 

CD :  rad. ::  BD :  sin.  BCD(=HDB), 
the  depression  of  the  horizon. 

The  depression  thus  obtained  is  the  true  depression ;  but  this 
must  be  lessened  by  the  amount  of  terrestrial  refraction,  which  is 
very  uncertain.  About  |th  or  -^th  of  the  whole  quantity  is  usu- 
ally allowed. 

For  an  elevation  of  25  feet  the  dip  of  the  horizon  amounts  to 
nearly  five  minutes,  and  for  an  elevation  of  100  feet  it  amounts 
to  nearly  ten  minutes.  For  an  elevation  of  8000  feet  the  dip 
amounts  to  about  one  degree,  and  for  an  elevation  of  12,000  feet 
it  amounts  to  about  two  degrees. 

In  1862  Mr.  Glaisher  ascended  in  a  balloon  to  the  height  of 
36,670  feet,  from  which  elevation  the  dip  of  the  horizon  was 
nearly  four  degrees. 


CHAPTER  III. 

ATMOSPHERIC   REFRACTION. — TWILIGHT. 

84.  THE  air  which  surrounds  the  earth  decreases  gradually  in 
density  as  we  ascend  from  the  surface.  At  the  height  of  4  miles, 
the  density  is  only  about  half  as  great  as  at  the  earth's  surface ; 
at  the  height  of  8  miles  about  one  fourth  as  great ;  at  the  height 
of  12  miles  about  one  eighth  as  great,  and  so  on.  From  this  law 
it  follows  that  at  the  height  of  50  miles,  its  density  must  be  ex- 
tremely small,  so  as  to  be  nearly  or  quite  insensible. 


52  ASTRONOMY. 

85.  Law  of  atmospheric  refraction.— According  to  a  law  of  optics, 
when  a  ray  of  light  passes  obliquely  from  a  rarer  to  a  denser  me- 
dium, it  is  bent  toward  the  perpendicular  to  the  refracting  sur- 
face. Let  SA  be  a  ray  of  light  coming 
from  any  distant  object,  S,  and  falling 
on  the  surface  of  a  series  of  layers  of 
air,  increasing  in  density  downward. 
The  ray  SA,  passing  into  the  first  lay- 
er, will  be  deflected  in  the  direction 
AB,  toward  a  perpendicular  to  the  sur- 


•    face,  MN.     Passing  into  the  next  lay- 
er, it  will  be  again  deflected  in  the  di- 


rection BC,  more  toward  the  perpen- 
dicular ;  and  passing  through  the  lowest  layer,  it  will  be  still  more 
deflected,  and  will  enter  the  eye  at  D,  in  the  direction  of  CD;  and, 
since  every  object  appears  in  the  direction  from  which  the  visual 
ray  enters  the  eye,  the  object  S  will  be  seen  in  the  direction  DS', 
instead  of  its  true  direction  AS. 

Since  the  density  of  the  earth's  atmosphere  increases  gradually 
from  its  upper  surface  to  the  earth,  when  a  ray  of  light  from  any 
of  the  heavenly  bodies  enters  the  atmosphere  obliquely,  its  path 
is  not  a  broken  line,  as  we  have  here  supposed,  but  a  curve,  con- 
cave toward  the  earth.  The  density  of  the  upper  parts  of  the  at- 
mosphere being  very  small,  the  curve  at  first  deviates  very  little 
from  a  straight  line,  but  the  deviation  increases  as  it  approaches 
the  earth.  Both  the  straight  and  curved  parts  of  the  ray  lie  in 
the  same  vertical  plane ;  that  is,  the  refraction  of  the  atmosphere 
makes  an  object  appear  to  be  nearer  the  zenith  than  it  really  is, 
but  does  not  affect  its  azimuth. 

86.  How  the  refraction  may  be  computed. — It  is  a  difficult  prob- 
lem to  compute  the  exact  amount  of  the  refraction  of  the  atmos- 
phere ;  but  for  altitudes  exceeding  10  degrees,  the  entire  refrac- 
tion may  be  assumed  to  take  place  at  a  single  surface,  as  MN,  and 
may  be  computed  approximately  in  the  following  manner: 

Let  2  denote  the  apparent  zenith  distance  of  a  star,  and  r  the 
effect  of  refraction ;  then,  if  there  were  no  refraction,  the  zenith, 
distance  would  be  z+r.  But  we  have  found  in  Optics,  Art.  691, 
that  the  sine  of  incidence  =  mx  sine  of  refraction,  where  m  repre- 
sents the  index  of  refraction.  Ilence 


ATMOSPHERIC   REFRACTION".  53 

sin.  (z+r)=m  sin.  2. 
But  by  Trigonometry,  Art.  72, 

sin.  (z+r)=sin.z  cos.  r-f  cos.  z  sin.  r. 

For  zenith  distances  less  than  80°,  r  is  less  than  6',  and  therefore 
its  sine  may  be  considered  equal  to  the  arc,  and  its  cosine  equal 
to  unity.  Hence  we  find 

sin.  z+r  cos.  z=m  sin.  z. 

Dividing  by  cos.  z,  and  putting  — '= tangent  (Trigonometry,  Art. 

28),  we  obtain  tang.  z±r=m  tang,  z, 

or  r  =  (m— 1)  tang.  z. 

r  is  here  expressed  in  parts  of  radius.     If  we  wish  to  have  its 
value  expressed  in  seconds,  we  must  multiply  it  by  206265,  which 
is  the  number  of  seconds  in  an  arc  equal  to  radius.     If  r"  repre- 
sents the  refraction  expressed  in  seconds,  then 
r"=206265  r. 

At  the  temperature  of  50°,  and  pressure  29.96  inches,  the  re- 
fractive index  of  air  is  1.0002836.  Hence  we  have 

r"  =  0.0002836  X  206265  x  tang.  z. 
or  r"=58".49,tang.z; 

that  is,  the  refraction  is  equal  to  58/r.49  x  tangent  of  the  zenith 
distance.  For  altitudes  greater  than  40°  the  error  of  this  formula 
does  not  exceed  one  second,  and  for  an  altitude  of  12°  the  error 
does  not  exceed  ten  seconds.  The  following  formula  is  much 
more  accurate,  viz.,  r"=57".54  tang,  (z— 3r"). 

For  altitudes  greater  than  12°  the  error  of  this  formula  never 
exceeds  a  half  second,  and  for  altitudes  as  small  as  6°  the  error 
never  amounts  to  ten  seconds ;  but  to  furnish  the  refraction  with 
accuracy  entirely  down  to  the  horizon  requires  a  much  more 
complicated  formula. 

From  a  comparison  of  an  immense  number  of  observations,  it 
has  been  found  that  the  average  value  of  the  refraction  at  the 
horizon  is  about  thirty-five  minutes,  or  a  little  more  than  half  a 
degree ;  at  an  altitude  of  10°  it  is  only  five  minutes ;  at  25°  it  is 
two  minutes;  at  45°  it  is  one  minute;  at  62°  it  is  only  thirty 
seconds ;  and  in  the  zenith  it  is  zero.  See  Table  VII.,  page  328. 

87.  How  the  refraction  may  be  determined  by  observation. — The 
amount  of  refraction  for  different  altitudes  may  be  determined  by 
observation  as  follows  :  In  latitudes  greater  than  45°,  a  star  which 


54  ASTRONOMY. 

passes  through  the  zenith  of  the  place,  may  also  be  observed  when 
it  passes  the  meridian  below  the  pole.  Let  the  polar  distance  of 
such  a  star  be  measured  both  at  the  upper  and  lower  culmina- 
tions. In  the  former  case  there  will  be  no  refraction ;  the  differ- 
ence between  the  two  observed  polar  distances  will  therefore  be 
the  amount  of  refraction  for  the  altitude  at  the  lower  culmination ; 
because  if  there  were  no  refraction,  the  apparent  diurnal  path  of 
the  star  would  be  a  circle  with  the  celestial  pole  for  its  centre. 
This  method  is  strictly  applicable  only  in  latitudes  greater  than 
45°,  and  by  observations  at  one  station  we  can  only  determine  the 
refraction  corresponding  to  a  single  altitude.  Since,  however, 
for  zenith  distances  less  than  45°,  the  amount  of  refraction  is  quite 
small,  and  is  given  with  great  accuracy  by  the  Tables,  we  may 
safely  extend  the  application  of  this  method.  We  may  therefore 
select  any  star  within  the  circle  of  perpetual  apparition,  and  ob- 
serve its  polar  distance  at  the  upper  and  lower  culminations,  and 
correct  the  former  for  refraction.  The  difference  between  this 
corrected  value  and  the  observed  polar  distance  at  the  lower  cul- 
mination, will  be  the  refraction  corresponding  to  the  latter  altitude. 

88.  Second  method  of  determining  refraction.  —  The  following 
method  is  more  general  in  its  application,  and  will  enable  us  to 
construct  a  complete  table  of  refractions. 

Observe  the  altitude  of  a  star  whose  declination  is  known,  and 
note  the  time  by  the  clock.  Observe  also  when  the  star  crosses 
the  meridian,  and  the  difference  of  time  between  the  observations 
will  give  the  hour  angle  of  the  star  from  the  meridian. 

Fig.  25.     z  Let  PZEE  be  the  meridian  of  the 

place  of  observation,  P  the  pole,  Z 
the  zenith,  and  S  the  true  place  of 
the  star.     Let  ZS  be  a  vertical  cir- 
cle passing  through  the  star,  and 
1 0  PS  an  hour  circle  passing  through 
the  star.     Then,  in  the  triangle  ZPS, 

PZ=the  complement  of  the  latitude, 
PS=the  north  polar  distance  of  the  star, 
and    ZPS = the  angular  distance  of  the  star  from  the  meridian. 
In  this  triangle  we  know,  therefore,  two  sides  and  the  included 
angle,  from  which  we  can  compute  ZS,  or  the  true  zenith  distance 
of  the  star.     The  difference  between  the  computed  value  of  ZS 


ATMOSPHERIC  REFRACTION-.  55 

and  its  observed  value,  will  be  the  refraction  corresponding  to  this 
altitude. 

If  we  commence  our  observations  when  the  star  is  near  the  ho- 
rizon, and  continue  them  at  short  intervals  until  it  reaches  the 
meridian,  we  may,  by  a  proper  selection  of  stars,  determine  the 
amount  of  refraction  for  all  altitudes  from  zero  to  90°. 

89.  Corrections  for  temperature  and  pressure. — The  amount  of 
refraction  at  a  given  altitude  is  not  constant,  but  depends  upon 
the  temperature,  and  weight  of  the  air.     Tables  have  been  con- 
structed, partly  from  observation  and  partly  from  theory,  by  which 
we  may  at  once  obtain  the  mean  refraction  for  any  altitude ;  and 
rules  are  given  by  which  a  correction  may  be  made  for  the  state 
of  the  barometer  and  thermometer. 

90.  Effect  of  refraction  upon  the  time  of  sunrise. — Since  refraction 
increases  the  altitudes  of  the  heavenly  bodies,  it  must  accelerate 
their  rising  and  retard  their  setting,  and  thus  render  them  longer 
visible.     The  amount  of  refraction  at  the  horizon  is  about  35', 
which  being  a  little  more  than  the  apparent  diameters  of  the  sun 
and  moon,  it  follows  that  these  bodies,  at  the  moment  of  rising 
and  setting,  are  visible  above  the  horizon,  when  in  reality  they 
are  wholly  below  it. 

91.  Effect  of  refraction  upon  the  figure  of  the  sun's  disc. — When 
the  sun  is  near  the  horizon,  the  lower  limb,  being  nearest  the  ho- 
rizon, is  most  affected  by  refraction,  and  therefore  more  elevated 
than  the  upper  limb,  the  effect  of  which  is  to  bring  the  two  limbs 
apparently  closer  together  by  the  difference  between  the  two  re- 
fractions.   The  apparent  diminution  of  the  vertical  diameter  some- 
times amounts  at  the  horizon  to  one  fifth  of  the  whole  diameter. 
The  disc  thus  assumes  the  form  of  an  ellipse,  of  which  the  major 
axis  is  horizontal. 

92.  Enlargement  of  the  sun  near  the  horizon. — The  apparent  en- 
largement of  the  sun  and  moon  near  the  horizon  is  an  optical  illu- 
sion.   If  we  measure  the  apparent  diameters  of  these  bodies  with 
any  suitable  instrument,  we  shall  find  that  they  subtend  a  less 
angle  near  the  horizon,  than  they  do  when  near  the  zenith.    It  is, 
then,  wholly  owing  to  an  error  of  judgment  that  they  seem  to  ua 
larger  near  the  horizon. 


56  ASTRONOMY. 

Our  judgment  of  the  absolute  magnitude  of  a  body  is  based 
upon  our  judgment  of  its  distance.  If  two  objects  at  unequal  dis- 
tances subtend  the  same  angle,  the  more  distant  one  must  be  the 
larger.  Now  the  sun  and  moon,  when  near  the  horizon,  appear  to 
us  °more  distant  than  when  they  are  high  in  the  heavens.  They 
seem  more  distant  in  the  former  position,  partly  from  the  number 
of  intervening  objects,  and  partly  from  diminished  brightness. 
When  the  moon  is  near  the  horizon,  a  variety  of  intervening  ob. 
jects  shows  us  that  the  distance  of  the  moon  must  be  considera- 
ble ;  but  when  the  moon  is  on  the  meridian  no  such  objects  inter- 
vene, and  the  moon  appears  quite  near.  For  the  same  reason,  the 
vault  of  heaven  does  not  present  the  appearance  of  a  hemisphere, 
but  appears  flattened  at  the  zenith,  and  spread  out  at  the  horizon. 

Our  estimate  of  the  distance  of  objects  is  also  affected  by  their 
brightness.  Thus,  a  distant  mountain,  seen  through  a  perfectly 
clear  atmosphere,  appears  much  nearer  than  when  seen  through 
a  hazy  atmosphere. 

93.  Cause  of  twilight. — The  sun  continues  to  illumine  the  clouds 
and  the  upper  strata  of  the  air,  after  it  has  set,  in  the  same  man- 
.  ner  as  it  shines  on  the  summits  of  mountains  after  it  has  set  to 
the  inhabitants  of  the  adjacent  plains.  The  air  and  clouds  thus 
illumined  reflect  light  to  the  earth  below  them,  and  produce  twi- 
light. As  the  sun  continues  to  descend  below  the  horizon,  a  less 
part  of  the  visible  atmosphere  receives  his  direct  light ;  less  light 
is  transmitted  by  reflection  to  the  surface  of  the  earth ;  until,  at 
length,  all  reflection  ceases,  and  night  begins.  This  takes  place 
when  the  sun  is  about  18°  below  the  horizon. 

Before  sunrise  in  the  morning,  the  same  phenomena  are  exhib- 
ited in  the  reverse  order.  If  there  were  no  atmosphere,  none  of 
the  sun's  rays  could  reach  us  after  his  actual  setting,  or  before  his 
rising. 

Let  ABCD  repre- 
sent a  portion  of  the 
earth,  A  a  point  on 
its  surface  where  the 
sun,  S,  is  in  the  act  of 
setting,  and  let  SAH 
be  a  ray  of  light  just 
grazing  the  earth  at 


TWILIGHT.  57 

A,  and  leaving  the  atmosphere  at  the  point  II.  The  point  A  is 
illuminated  by  the  whole  reflective  atmosphere  HGFE.  The 
point  B,  to  which  the  sun  has  set,  receives  no  direct  solar  light, 
nor  any  reflected  from  that  part  of  the  atmosphere  which  is  be- 
low ALH,  but  it  receives  a  twilight  from  the  portion  ELF,  which 
lies  above  the  visible  horizon  BF.  The  point  C  receives  a  twi- 
light only  from  the  small  portion  of  the  atmosphere  HMGr,  while 
at  D  the  twilight  has  ceased  altogether. 

94.  Duration  of  twilight  at  the  equator. — The  duration  of  twilight 
varies  with  the  season  of  the  year,  and  with  our  position  upon  the 
earth's  surface.    At  the  equator,  where  the  circles  of  daily  rotation, 
are  perpendicular  to  the  horizon,  when  the  sun  is  in  the  celestial 
equator,  it  descends  through  18°  in  an  hour  and  twelve  minutes 
(-}4i=li  hours) ;  that  is,  twilight  lasts  Ih.  and  12m.    When  the  sun 
is  not  in  the  equator,  the  duration  of  twilight  is  somewhat  increased. 

95.  Duration  of  twilight  at  the  poles. — At  the  north  pole  there  is 
night  as  long  as  the  sun  is  south  of  the  equator;  but  whenever  it 
is  not  more  than  18°  south,  the  sun  is  never  more  than  18°  below 
the  horizon.     About  the  close  of  September,  the  sun  sinks  below 
the  horizon,  and  there  is  continual  twilight  until  November  12th, 
when  it  attains  a  distance  of  18°  from  the  equator.     From  this 
date  there  is  no  twilight  until  January  29th,  from  which  time  there 
is  continual  twilight  until  about  the  middle  of  March,  when  the 
sun  rises  above  the  horizon,  and  continues  above  the  horizon  un- 
interruptedly for  six  months. 

96.  Duration  of  twilight  in  middle  latitudes. — At  intermediate 
points  of  the  earth,  the  duration  of  twilight  may  vary  from  Ih. 
12m.  to  several  weeks.     In  latitude  40°,  during  the  months  of 
March  and  September,  twilight  lasts  about  an  hour  and  a  half, 
while  in  midsummer  it  lasts  a  little  over  two  hours. 

In  latitude  50°,  where  the  north  pole  is  elevated  50°  above  the 
horizon,  the  point  which  is  on  the  meridian  18°  below  the  north 
point  of  the  horizon,  is  68°  distant  from  the  north  pole,  and  there- 
fore 22°  distant  from  the  equator.  Now,  during  the  entire  month 
of  June,  the  distance  of  the  sun  from  the  equator  exceeds  22° ; 
that  is,  in  latitude  50°  there  is  continual  twilight  from  sunset  to 
sunrise,  during  a  period  of  more  than  a  month. 


58  ASTRONOMY. 

At  places  nearer  to  the  pole,  the  period  of  the  year  during  which 
twilight  lasts  through  the  entire  night,  is  still  longer. 

97.  Consequences  if  there  tvere  no  atmosphere. — If  there  were  no 
atmosphere,  the  darkness  of  midnight  would  instantly  succeed  the 
setting  of  the  sun,  and  it  would  continue  thus  until  the  instant  of 
the  sun's  rising.  During  the  day  the  illumination  would  also  be 
much  less  than  it  is  at  present,  for  the  sun's  light  could  only  pene- 
trate apartments  which  were  directly  accessible  to  his  rays,  or  into 
which  it  was  reflected  from  the  surface  of  natural  objects.  On 
the  summits  of  mountains,  where  the  atmosphere  is  very  rare,  the 
sky  assumes  the  color  of  the  deepest  blue,  approaching  to  black- 
ness, and  stars  become  visible  in  the  daytime. 


CHAPTER  IV. 
THE  EARTH'S  ANNUAL  MOTION. — SIDEREAL  AND  SOLAR  TIME. — 

THE  EQUATION  OF  TIME.  — THE   CALENDAR.  —  THE  CELESTIAL 
GLOBE. 

98.  Sun's  apparent  motion  in  right  ascension. — If  we  observe  the 
exact  position  of  the  sun  with  reference  to  the  stars,  from  day  to 
day  through  the  year,  we  shall  find  that  it  has  an  apparent  mo- 
tion among  them  along  a  great  circle  of  the  celestial  sphere,  whose 
plane  makes  an  angle  of  23°  27'  with  the  plane  of  the  celestial 
equator.     This  motion  may  be  determined  by  observations  with 
the  transit  instrument  and  mural  circle. 

If  the  sun's  transit  be  observed  daily,  and  its  right  ascension  be 
determined,  it  will  be  found  that  the  right  ascension  increases  each 
day  about  four  minutes  of  time,  or  one  degree,  so  that  in  a  year 
the  sun  makes  a  complete  circuit  round  the  heavens,  moving  con- 
stantly among  the  stars  from  west  to  east.  This  daily  motion  in 
right  ascension  is  not  uniform,  but  varies  from  215s.  to  266s.,  the 
mean  being  about  236s.,  or  3m.  56s. 

99.  Sun's  apparent  motion  in  declination. — If  the  point  at  which 
the  sun's  centre  crosses  the  meridian  be  observed  daily  with  the 
mural  circle,  it  will  be  found  to  change  from  day  to  day.    Its 
declination  is  zero  on  the  20th  of  March,  from  which  time  its  north 


THE  EARTH'S  AXXUAL  MOTION.  59 

declination  increases  until  it  becomes  23°  27'  on  the  21st  of  June. 
It  then  decreases  until  the  22d  of  September,  when  the  sun's  cen- 
tre is  again  upon  the  equator.  Its  south  declination  then  increases 
until  it  becomes  23°  27'  on  the  21st  of  December,  after  which  it 
decreases  until  the  sun's  centre  returns  to  the  equator  on  the  20th 
of  March. 

If  we  trace  upon  a  celestial  globe  the  course  of  the  sun  from 
day  to  day,  we  shall  find  its  path  to  be  a  great  circle  of  the  heav- 
ens, inclined  to  the  equator  at  an  angle  of  23°  27'.  This  circle  is 
called  the  ecliptic,  because  solar  and  lunar  eclipses  can  only  take 
place  when  the  moon  is  very  near  this  plane. 

100.  The  equinoxes  and  solstices. — The  ecliptic  intersects  the  ce- 
lestial equator  at  two  points  diametrically  opposite  to  each  other. 
These  are  called  the  equinoctial  points  •  because,  when  the  sun  is 
at  these  points,  it  is  for  an  equal  time  above  and  below  the  hori- 
zon, and  the  days  and  nights  are  therefore  equal. 

The  point  at  which  the  sun  passes  from  the  south  to  the  north 
side  of  the  celestial  equator,  is  called  the  vernal  equinoctial  point, 
and  the  other  is  called  the  autumnal  equinoctial  point.  The  times 
at  which  the  sun's  centre  is  found  at  these  points  are  called  the 
vernal  and  autumnal  equinoxes.  The  vernal  equinox,  therefore, 
takes  place  on  the  20th  of  March,  and  the  autumnal  on  the  22d 
of  September. 

Those  points  of  the  ecliptic  which  are  midway  between  the 
equinoctial  points  are  the  most  distant  from  the  celestial  equator, 
and  are  called  the  solstitial  points;  and  the  times  at  which  the 
sun's  centre  passes  those  points  are  called  the  solstices.  The  sum- 
mer solstice  takes  place  on  the  21st  of  June,  and  the  winter  sol- 
stice on  the  21st  of  December. 

101.  The  equinoctial  colure  is  the  hour  circle  which  passes 
through  the  equinoctial  points.     The  solstitial  colure  is  the  hour 
circle  which  passes  through  the  solstitial  points.     The  solstitial 
colure  is  at  right  angles  both  to  the  ecliptic  and  to  the  equator, 
for  it  cuts  both  these  circles  90  degrees  from  their  common  inter- 
section ;  that  is,  from  the  equinoctial  points. 

The  distance  of  either  solstitial  point  from  the  celestial  equator 
is  23°  27'.  The  more  distant  the  sun  is  from  the  celestial  equa- 
tor, the  more  unequal  will  be  the  days  and  nights ;  and,  therefore, 


60  ASTRONOMY. 

the  longest  day  of  the  year  will  be  the  day  of  the  summer  sol- 
stice, and  the  shortest  that  of  the  winter  solstice.  In  southern  lat- 
itudes the  seasons  will  be  reversed. 

102.  The  zodiac  is  a  zone  of  the  heavens  extending  eight  de- 
grees each  side  of  the  ecliptic.  The  sun,  the  moon,  and  all  the 
principal  planets,  have  their  motions  within  the  limits  of  the  zo- 
diac. 

The  zodiac  is  divided  into  twelve  equal  parts,  called  signs,  each 
of  which  contains  30  degrees.  Beginning  with  the  vernal  equi- 
nox, they  are  as  follows : 


Sign. 

Symbol. 

Sign. 

Symbol. 

I.  Aries. 
II.  Taurus. 
III.  Gemini. 
IV.  Cancer. 
V.  Leo. 
VI.  Virgo. 

T 
8 

n 

5> 

a 
nj? 

VII.  Libra. 
VIII.  Scorpio. 
IX.  Sagittarius. 
X.  Capricornus. 
XI.  Aquarius. 
XII.  Pisces. 

=£: 

TTl 
/ 

V3 
X 

The  vernal  equinox  is  at  the  first  point  of  Aries,  and  the  au- 
tumnal equinox  at  the  first  of  Libra.  The  summer  solstice  is  at 
the  first  of  Cancer,  and  the  winter  solstice  at  the  first  of  Capricorn. 

103.  The  tropics  are  two  small  circles  parallel  to  the  equator, 
and  passing  through  the  solstices.     That  on  the  north  of  the 
equator  is  called  the  Tropic  of  Cancer,  and  that  on  the  south  the 
Tropic  of  Capricorn. 

The  Polar  circles  are  two  small  circles  parallel  to  the  equator, 
and  distant  23°  27'  from  the  poles.  One  is  called  the  Arctic  and 
the  other  the  Antarctic  circle. 

A  great  circle  of  the  celestial  sphere  passing  through  the  poles 
of  the  ecliptic,  is  called  a  circle  of  latitude. 

The  latitude  of  a  star  is  the  distance  of  the  star  from  the  eclip 
tic,  measured  on  a  circle  of  latitude.  It  may  be  north  or  south, 
and  is  counted  from  zero  to  90  degrees. 

The  longitude  of  a  star  is  the  distance  from  the  vernal  equinox 
to  the  circle  of  latitude  passing  through  the  star,  measured  on  the 
ecliptic  in  the  order  of  the  signs.  Longitude  is  counted  from 
zero  to  360  degrees. 

104.  Appearances  produced  by  the  earth's  annual  motion.—  The 


THE  EARTH'S  ANNUAL  MOTION.  61 

apparent  annual  motion  of  the  sun  may  be  explained  either  by 
supposing  a  real  revolution  of  the  sun  around  the  earth,  or  a  rev- 
olution of  the  earth  around  the  sun.  But  it  follows  from  the  prin- 
ciples of  Mechanics  that  the  earth  and  sun  must  both  revolve 
around  their  common  centre  of  gravity,  and  this  point  is  very 
near  the  centre  of  the  sun. 

If  the  earth  could  be  observed  by  a  spectator  upon  the  sun,  it 
would  appear  among  the  fixed  stars  in  the  point  of  the  sky  oppo- 
site to  that  in  which  the  sun  appears  as  viewed  from  the  earth. 

Thus,  in  Fig.  27,  let  S  represent 
the  sun,  and  ABPD  the  earth's 
orbit :  a  spectator  upon  the  earth 
will  see  the  sun  projected  among 
the  fixed  stars  in  the  point  of  the 
sky  opposite  to  that  occupied  by 
the  earth ;  and,  as  the  earth  moves 
from  A  to  B  and  P,  the  sun  will 
appear  to  move  among  the  stars  ^  " 
from  P  to  D  and  A,  and  in  the 
course  of  the  year  will  appear  to 
trace  out  in  the  sky  the  plane  of  the  ecliptic.  When  the  earth  is 
in  Libra  we  see  the  sun  in  the  opposite  sign  Aries;  and  as  the 
earth  moves  from  Libra  to  Scorpio,  the  sun  appears  to  move  from 
Aries  to  Taurus,  and  so  on  through  the  ecliptic. 

105.  Phenomena  tuithin  the  arctic  circle. — At  the  summer  sol- 
stice, on  the  arctic  circle,  the  sun's  distance  from  the  north  pole  is 
just  equal  to  the  latitude  of  the  place,  and  the  sun's  diurnal  path 
just  touches  the  horizon  at  the  north  point.     Within  the  arctic 
circle,  there  will  be  several  days  during  which  the  sun  never  sinks 
below  the  horizon.     So,  also,  near  the  winter  solstice,  within  the 
arctic  circle,  there  will  be  several  days  during  which  the  sun  does 
not  rise  above  the  horizon. 

106.  Division  of  the  earth  into  zones. — The  earth  is  naturally  di- 
vided into  five  zones,  depending  on  the  appearance  of  the  diurnal 
path  of  the  sun. 

These  zones  are, 

1st.  The  two  frigid  zones,  included  within  the  polar  circles. 
Within  these  zones  there  are  several  days  of  the  year  during 


62 


ASTRONOMY. 


which  the  sun  does  not  rise  above  the  horizon,  and  other  days 
during  which  the  sun  does  not  sink  below  the  horizon. 

2d.  The  torrid  zone,  extending  from  the  Tropic  of  Cancer  to 
the  Tropic  of  Capricorn.  Throughout  this  zone,  the  sun  every 
year  passes  through  the  zenith  of  the  observer,  when  the  sun's 
declination  is  equal  to  the  latitude  of  the  place. 

3d.  The  north  and  south  temperate  zones,  extending  from  the 
tropics  to  the  polar  circles.  Within  these  zones  the  sun  is  never 
seen  in  the  zenith,  and  it  rises  and  sets  every  day. 

107.  Cause  of  the  change  of  seasons. — "While  the  earth  revolves 
annually  round  the  sun,  it  has  a  motion  of  rotation  upon  an  axis 
which  is  inclined  23°  27'  from  a  perpendicular  to  the  ecliptic; 
and  tiiis  axis  continually  points  in  the  same  direction.  Hence  result 
the  alternations  of  day  and  night,  and  the  succession  of  seasons. 

In  June,  when  the  north  pole  of  the  earth  inclines  toward  the 
sun,  the  greater  portion  of  the  northern  hemisphere  is  enlightened, 
and  the  greater  portion  of  the  southern  hemisphere  is  dark.  The 
days  are,  therefore,  longer  than  the  nights  in  the  northern  heini- 


Fig.  28. 


THE  EAKTH'S  ANNUAL  MOTION.  63 

sphere.  The  reverse  is  true  in  the  southern  hemisphere ;  but  on 
the  equator,  the  days  and  nights  are  equal.  In  December,  when 
the  south  pole  inclines  toward  the  sun,  the  days  are  longer  than 
the  nights  in  the  southern  hemisphere. 

In  March  and  September,  when  the  earth's  axis  is  perpendicu- 
lar to  the  direction  of  the  sun,  the  circle  which  separates  the  en- 
lightened from  the  unenlightened  hemisphere,  passes  through  the 
poles,  and  the  days  and  nights  are  equal  all  over  the  globe. 

These  different  cases  are  illustrated  by  Fig.  28.  Let  S  represent 
the  position  of  the  sun,  and  ABCD  different  positions  of  the  earth 
in  its  orbit,  the  axis  ns  always  pointing  toward  the  same  fixed 
star.  At  A  and  C  the  sun  illumines  from  n  to  s,  and  as  the  globe 
turns  upon  its  axis,  the  sun  will  appear  to  describe  the  equator, 
and  the  days  and  nights  will  be  equal  in  all  parts  of  the  globe. 
When  the  earth  is  at  B,  the  sun  illumines  23-^°  beyond  the  north 
pole  n,  and  falls  the  same  distance  short  of  the  south  pole  s.  When 
the  earth  is  at  D,  the  sun  illumines  23-2-°  beyond  the  south  pole  s, 
and  falls  the  same  distance  short  of  the  north  pole  n. 

-108.  Under  what  circumstances  would  there  have  been  no  change 
of  seasons  ? — If  the  earth's  axis  had  been  perpendicular  to  the 
plane  of  its  orbit,  the  equator  would  have  coincided  with  the  eclip- 
tic ;  day  and  night  would  have  been  of  equal  duration  throughout 
the  year,  and  there  would  have  been  no  diversity  of  seasons. 

109.  In  what  case  would  the  change  of  seasons  have  been  greater 
than  it  now  -is  ? — If  the  inclination  of  the  equator  to  the  ecliptic 
had  been  greater  than  it  is,  the  sun  would  have  receded  farther 
from  the  equator  on  the  north  side  in  summer,  and  on  the  south 
side  in  winter ;  and  the  heat  of  summer,  as  well  as  the  cold  of 
winter,  would  have  been  more  intense  ;  that  is,  the  diversity  of 
the  seasons  would  have  been  greater  than  it  is  at  present  If  the 
equator  had  been  at  right  angles  to  the  ecliptic,  the  poles  of  the 
equator  would  have  been  situated  in  the  ecliptic ;  and  at  the  sum- 
mer solstice  the  sun  would  have  appeared  at  the  north  pole  of  the 
celestial  sphere,  and  at  the  winter  solstice  it  would  have  been  at 
the  south  pole  of  the  celestial  sphere.  To  an  observer  in  the  mid- 
dle latitudes,  the  sun  would  therefore,  for  a  considerable  part  of 
summer,  be  within  the  circle  of  perpetual  apparition,  and  for  sev- 
eral weeks  be  constantly  above  the  horizon.  So,  also,  for  a  con- 


64  ASTRONOMY. 

siderable  part  of  winter,  he  would  be  within  the  circle  of  perpet- 
ual occultation,  and  for  several  weeks  be  constantly  below  the  ho- 
rizon. The  great  vicissitudes  of  heat  and  cold  resulting  from  such 
a  movement  of  the  sun,  would  be  extremely  unfavorable  to  both 
animal  and  vegetable  life. 

110.  To  determine  the  obliquity  of  the  ecliptic. — The  inclination  of 
the  equator  to  the  ecliptic,  or  the  obliquity  of  the  ecliptic,  is  equal 
to  the  sun's  greatest  declination.    It  may  therefore  be  ascertained 
by  measuring,  by  means  of  the  mural  circle,  the  sun's  declination 
at  the  summer,  or  at  the  winter  solstice.     The  greatest  declina- 
tion of  the  sun  is  found  to  be  23°  27'  25",  both  north  and  south 
of  the  equator.    This  arc  is,  however,  diminishing  at  the  rate  of 
about  half  a  second  annually. 

111.  Form  of  the  earths  orbit. — The  path  of  the  earth  around  the 
sun  is  nearly,  but  not  exactly,  a  circle.    The  relative  distances  of 
the  sun  from  the  earth  may  be  found  by  observing  the  changes 
in  the  sun's  apparent  diameter.     The  apparent  diameter  of  the 
sun,  at  different  distances  from  the  spectator,  varies  inversely  as 

F5g.  29.  the  distance.     Thus,  in  Fig.  29, 

E:sin.E::ES:AS. 

E        •     v     AS  .  1 

v  sin.  L— -jT-r,  or  varies  as  ^-r-r. 

JliO  ilife 

Since  the  sines  of  small  angles 
are  nearly  proportional  to  the  angles,  E  varies  as  ^-r,  very  nearly. 

J-JO 

By  measuring,  therefore,  the  sun's  apparent  diameter  from  day 
to  day  throughout  the  year,  we  have  the  means  of  determining 
the  relative  distances  of  the  sun  from  the  earth. 

Ex.  1.  On  the  1st  of  January,  1864,  the  sun's  apparent  diameter 
was  32'  36".4,  and  on  the  1st  of  July,  1864,  his  diameter  was 
31'  31".8.  Find  the  relative  distances  of  the  sun  at  these  two 
Periods.  A ns.  0.96698. 

Ex.  2.  On  the  1st  of  April,  1864,  the  apparent  diameter  of  the 
sun  was  32'  3".4.  Find  the  ratio  of  its  distance  to  the  distances 
in  July  and  January.  Ans.  0.98357  and  1.01716. 

112.  The  earths  orlitis  an  ellipse.— By  observations  of  the  sun's 
apparent  diameter  continued  throughout  the  year,  we  find  that 


THE  EARTH'S  ANNUAL  MOTION.  65 

the  true  form  of  the  earth's  orbit  is  an  ellipse,  having  the  sun  in 
one  of  the  foci.  The  sun's  apparent  diameter  is  least  on  the  1st 
of  July,  and  greatest  on  the  1st  of  January.  We  may  then  con- 
struct a  figure  showing  the  form  of  the  orbit,  by  setting  off  lines, 
SA,  SB,  SO,  etc.,  corresponding  to  Fig.so. 

the  sun's  distances,  and  making  an- 
gles with  each  other  equal  to  the 
sun's  angular  motion  between  the 
times  of  observation.  The  figure 
thus  formed  is  found  to  be  an  ellipse,  G' 
with  the  sun  occupying  one  of  the 
foci,  as  S. 

113.  To  find  the  eccentricity  of  the 
earth's  orbit. — The  point  A  of  the  orbit  where  the  earth  is  nearest 
the  sun,  is  called  the  perihelion,  and  this  happens  on  the  1st  of 
January.  The  point  Gr  most  distant  from  the  sun  is  called  its 
aphelion,  and  this  happens  on  the  1st  of  July ;  that  is,  the  earth 
is  more  distant  from  the  sun  in  summer  than  in  winter. 

The  distance  from  the  centre  of  the  ellipse  to  the  focus,  divided 
by  the  semi-major  axis,  is  called  the  eccentricity  of  the  ellipse,  and 
its  value  may  be  determined  as  follows : 

If  a  denote  the  semi-major  axis,  and  e  the  eccentricity  of  the 
earth's  orbit,  then 

the  earth's  aphelion  distance    =a(l  +  e); 
the  earth's  perihelion  distance  =  a(l—e). 

If  we  represent  the  aphelion  distance  by  A,  and  the  perihelion 
distance  by  P,  we  have 

A  =  l  +  e 

Solving  this  equation,  we  obtain 

A-P_   ~A 

"A+p-^~P" 

But  |=|^|=o.96698. 

A     o2  ob  A 

Hence  e= 0.01678, 

which  is  about  -j^-th. 

This  eccentricity  is  subject  to  a  diminution  of  0.000042  in  one 
E 


66  ASTRONOMY. 

hundred  years.  If  this  change  were  to  continue  indefinitely,  the 
earth's  orbit  must  eventually  become  circular ;  but  Le  Verrier  has 
proved  that  the  diminution  is  not  to  continue  beyond  24,000  years, 
when  the  eccentricity  will  be  equal  to  .0033,  and  after  that  time 
the  eccentricity  will  increase. 

114.  Law  of  Hie  earth's  motion  in  its  orbit.— The  radius  vector  of 
the  earth's  orbit  describes  equal  areas  in  equal  times.    Let  A  and 
FIg  3t  B  be  the  positions  of  the  earth  in  its 

orbit  on  two  successive  days ;  let  0  rep- 
resent the  angle  ASB,  and  R  represent 
AS.  Draw  AC  perpendicular  to  SB. 
Then  AC  =  AS  sin.  ASB  =  R  sin.  0 ; 
and  the  area  ASB=£R2  sin.  0.  But 
since  the  earth's  diurnal  motion  in  the 
ecliptic  is  small,  we  may  assume  that 
the  arc  0  is  equal  to  its  sine,  and  hence 
the  area=£R20. 

If  this  area  described  by  the  radius  vector  in  one  day,  is  a  con> 
stant  quantity,  then  R20  will  evidently  be  a  constant  quantity. 
But  R  varies  inversely  as  the  apparent  diameter  of  the  sun. 

n 

Hence,  putting  D  for  the  sun's  apparent  diameter,  =—  must  be  a 

constant  quantity ;  or 

0:0'::D2:D'2; 

that  is,  the  sun's  diurnal  motion  in  different  parts  of  its  orbit,  must 
vary  as  the  square  of  its  apparent  diameter. 

Now  we  find  this  supposition  verified  by  observation.     Thus : 

From  noon  of  January  1st  to  noon  of  January  2d,  1864,  the 
sun  moved  through  1°  1'  9".9  of  the  ecliptic ;  and  his  apparent 
diameter  at  the  same  time  was  32'  36".4. 

From  noon  of  July  1st  to  noon  of  July  2d,  1864,  the  sun  moved 
through  57'  12 ".9 ;  and  his  apparent  diameter  at  the  same  time 
was  31'  31".8. 

Reducing  these  values  to  seconds,  we  have 

3669.9  :  3432.9 ::  1956.42 :  1891.82. 

"We  find  the  same  law  to  hold  true  in  other  parts  of  the  orbit, 
and  hence  it  is  considered  as  established  by  observation,  that  the 
radius  vector  of  the  earth's  orbit  describes  equal  areas  in  equal  times. 


SIDEREAL  AND  SOLAE  TIME.  67 

115.  Why  the  greatest  heat  and  cold  do  not  occur  at  the  solstices. — 
The  influence  of  the  sun  in  heating  a  portion  of  the  earth's  sur- 
face depends  upon  its  altitude  above  the*  horizon,  and  upon  the 
length  of  time  during  which  it  continues  above  the  horizon.    The 
greater  the  altitude,  the  less  obliquely  will  the  rays  strike  the  sur- 
face of  the  earth  at  noon,  and  the  greater  will  be  their  heating 
power.     Both  these  causes  conspire  to  produce  the  increased  heat 
of  summer,  and  the  diminished  heat  of  winter.     It  might  be  in- 
ferred that  the  hottest  day  ought  to  occur  on  the  21st  of  June, 
when  the  sun  rises  highest,  and  the  days  are  the  longest.     Such, 
however,  is  not  the  case,  for  the  following  reason :  As  midsum- 
mer approaches,  the  quantity  of  heat  imparted  by  the  sun  during 
the  day  is  greater  than  the  quantity  lost  during  the  night,  and 
hence  each  day  there  is  an  increase  of  heat.    On  the  21st  of  June 
this  daily  augmentation  reaches  its  maximum ;  but  there  is  still 
each  day  an  accession  of  heat,  until  the  heat  lost  during  the  night  is 
just  equal  to  that  imparted  during  the  day,  which  happens,  at  most 
places  in  the  northern  hemisphere,  some  time  in  July  or  August. 

For  the  same  reason,  the  greatest  cold  does  not  occur  on  the 
21st  of  December,  but  some  time  in  January  or  February. 

Sidereal  and  /Solar  Time. 

116.  Sidereal  Time. — The  interval  between  two  successive  re- 
turns of  the  vernal  equinox  to  the  same  meridian,  is  called  a  side- 
real day.     This  interval  represents  the  time  of  the  rotation  of  the 
earth  upon  its  axis,  and  is  not  only  invariable  from  one  month  to 
another,  but  has  not  changed  so  much  as  the  hundredth  part  of  a 
second,  in  two  thousand  years. 

117.  Solar  Time. — The  interval  between  two  successive  returns 
of  the  sun  to  the  same  meridian,  is  called  a  solar  day. 

The  sun  passes  through  860  degrees  of  longitude  in  one  year, 
or  365  days  5  hours  48  minutes  and  47.8  seconds;  so  that  the 
sun's  mean  daily  motion  in  longitude  is  found  by  the  proportion 

one  year :  one  day  : :  360°  :  daily  motion =59'  8".33. 
This  motion  is  not  uniform,  but  is  greatest  when  the  sun  is  near- 
est the  earth.  Hence  the  solar  days  are  unequal ;  and  to  avoid 
the  inconvenience  which  would  result  from  this  fact,  astronomers 
have  recourse  to  a  mean  solar  day,  the  length  of  which  is  equal  to 
the  mean  or  average  of  all  the  apparent  solar  days  in  a  year. 


63  ASTRONOMY. 

118.  Sidereal  and  solar  time  compared— -The  length  of  the  mean 
solar  day  is  greater  than  that  of  the  sidereal,  because  when  the 
mean  sun,  in  its  diurnal  motion,  returns  to  a  given  meridian,  it  is 
59'  8".3  eastward  of  its  position  on  the  preceding  day. 

An  arc  of  the  equator,  equal  to  360°  59'  8".3,  passes  the  merid- 
ian in  a  mean  solar  day,  while  only  360°  pass  in  a  sidereal  day. 
To  find  the  excess  of  the  solar  day  above  the  sidereal  day,  ex- 
pressed in  sidereal  time,  we  have  the  proportion 

360°  :  59'  8".3  : :  one  day  :  3m.  56.5s. 

Hence  24  hours  of  mean  solar  time  are  equivalent  to  24h.  3m. 
56.5s.  of  sidereal  time. 

To  find  the  excess  of  the  solar  day  above  the  sidereal  day,  ex- 
pressed in  solar  time,  we  have  the  proportion 

360°  59'  8".3  :  59'  8".3  : :  one  day  :  3m.  55.9s. 
Hence  24  hours  of  sidereal  time  are  equivalent  to  23h.  56m.  4.1s. 
of  mean  solar  time. 

119.  Civil  day,  and  astronomical  day. — The  civil  day  begins  at 
midnight,  and  consists  of  two  periods  of  12  hours  each  ;  but  mod- 
ern astronomers  commence  their  day  at  noon,  because  this  is  a 
date  which  is  marked  by  a  phenomenon  which  can  be  accurately 
observed,  viz.,  the  passage  of  the  sun  over  the  meridian  ;  and  be- 
cause observations  being  chiefly  made  at  night,  it  is  inconvenient 
to  have  a  change  of  date  at  midnight.     Tfie  astronomical  day 
commences  12  hours  later  than  the  civil  day,  and  the  hours  are 
numbered  continuously  up  to  24.     Thus  July  4th,  9  A.M.  civil 
time,  corresponds  to  July  3d,  21  hours  of  astronomical  time. 

120.  Apparent  time,  and  mean  time. — The  interval  between  two 
successive  returns  of  the  sun  to  the  same  meridian,  is  an  apparent 
solar  day ;  and  apparent  time  is  time  reckoned  in  apparent  solar 
days,  while  mean  time  is  time  reckoned  in  mean  solar  days. 

The  difference  between  apparent  solar  time  and  mean  solar 
time,  is  called  the  equation  of  time. 

If  a  clock  were  required  to  keep  apparent  solar  time,  it  would 
be  necessary  that  its  rate  should  change  from  day  to  day  accord" 
ing  to  a  complicated  law.  It  has  been  found  in  practice  impossi- 
ble to  accomplish  this,  and  hence  clocks  are  now  regulated  to  in- 
dicate mean  solar  time.  A  clock,  therefore,  should  not  indicate 
12h.  when  the  sun  is  on  the  meridian,  but  should  sometimes  indi- 


CAUSE   OF   INEQUALITY   IN   SOLAR  DAYS.  69 

cate  more  than  12h.  and  sometimes  less  than  12h.,  the  difference 
being  equal  to  the  equation  of  time. 

121.  Cause  of  the  inequality  of  the  solar  days. — The  inequality  of 
the  solar  days  depends  on  two  causes,  the  unequal  motion  of  the 
earth  in  its  orbit,  and  the  inclination  of  the  equator  to  the  ecliptic. 

While  the  earth  is  revolving  round  the  sun  in  an  elliptical  or- 
bit, its  motion  is  greatest  when  it  is  Fig.  32. 
nearest  the  sun,  and  slowest  when  it 
is  most  distant.     Let  ADGK  repre- 
sent the  elliptic  orbit  of  the  earth, 
with  the  sun  in  one  of  its  foci  at  S, 
and  let  the  direction  of  motion  be 
from  A  toward  E. 

We  have  found  that  the  sun's 
mean  daily  motion  as  seen  from  the 
earth,  or  the  earth's  mean  daily  mo- 
tion  as  seen  from  the  sun,  is  59'  8".3.  But  when  the  earth  is 
nearest  the  sun  its  daily  motion  is  61'  10".  In  passing  from  A 
toward  E  its  daily  motion  diminishes,  and  at  G  it  is  only  57'  12". 
While  moving,  therefore,  from  A  through  E  to  G,  the  earth  will 
be  in  advance  of  its  mean  place,  while  at  G,  having  completed  a 
half  revolution,  the  true  and  the  mean  places  will  coincide.  For 
a  like  reason,  in  going  from  G  to  A,  the  earth  will  be  behind  its 
mean  place ;  but  at  A  the  mean  and  true  places  will  again  coin- 
cide. This  point  A  in  the  diagram,  corresponds  to  about  the  1st 
of  January. 

Now  the  apparent  direction  of  the  sun  from  the  earth,  is  exact- 
ly opposite  to  that  of  the  earth  from  the  sun.  Hence,  when  the 
earth  is  nearest  to  the  sun,  the  apparent  solar  day  will  be  longer 
than  the  mean  solar  day.  If,  then,  we  conceive  a  fictitious  sun  to 
move  uniformly  through  the  heavens,  describing  59'  8"  per  day, 
and  that  the  true  and  fictitious  suns  are  together  on  the  1st  of 
January,  it  is  evident  that  on  the  2d  of  January  the  fictitious  sun 
will  come  to  the  meridian  a  few  seconds  before  the  true  sun ;  on 
the  3d  of  January  the  fictitious  sun  will  be  still  more  in  advance 
of  the  true  sun,  and  this  difference  will  go  on  increasing  for  about 
three  months,  when  it  amounts  to  a  little  more  than  8  minutes. 
From  this  time  the  difference  will  diminish  until  about  the  1st  of 
July,  when  the  positions  of  the  true  and  fictitious  suns  will  coin- 


70  ASTRONOMY. 

cide.  But  on  the  2d  of  July  the  fictitious  sun  will  come  to  the 
meridian  a  few  seconds  later  than  the  true  sun  ;  on  the  3d  of  July 
it  will  have  fallen  still  more  behind  the  true  sun,  and  this  differ- 
ence will  go  on  increasing  for  about  three  months,  when  it  amounts 
to  a  little  more  than  8  minutes.  From  this  time  the  difference 
will  diminish  until  the  1st  of  January,  when  the  positions  of  the 
true  and  fictitious  suns  will  again  coincide. 

So  far,  then,  as  it  depends  upon  the  unequal  motion  of  the 
earth  in  its  orbit,  the  equation  of  time  is  positive  for  six  months, 
and  then  negative  for  six  months,  and  its  greatest  value  is  8m.  2-is. 

122.  Second  cause  for  the  inequality  of  the  solar  days.  —  Even  if 
the  earth's  motion  in  its  orbit  were  perfectly  uniform,  the  appar- 
ent solar  days  would  be  unequal,  because  the  ecliptic  is  inclined  to 


Fig.  38. 


the  equator.  Let 
represent  the  equator, 
and  AGN  the  northern 
half  of  the  ecliptic.  Let 
the  ecliptic  be  divided 
into  equal  portions,  AB, 
BC,  CD,  etc.,  supposed 
to  be  described  by  the 
sun  in  equal  portions  of 
time  ;  and  through  the  points  B,  C,  D,  etc.,  let  hour  circles  be  made 
to  pass,  cutting  the  equator  in  the  points  Z>,  c,  d,  etc.  The  arc  AGN 
is  equal  to  the  arc  A^/N,  for  all  great  circles  bisect  each  other  ; 
also  AG  is  equal  to  Ag,  since  the  former  is  one  half  of  AGN,  and 
the  latter  of  A#N.  Now,  since  AB6  is  a  right-angled  triangle, 
AB  is  greater  than  Al  ;  for  the  same  reason,  AC  is  greater  than 
Ac;  AD  is  greater  than  AcZ,  and  so  on.  But  AG  is  equal  to  Ag  ; 
therefore  Ag  is  divided  into  unequal  portions  at  the  points  6,  c,  d, 
etc.  Now  B  and  b  come  to  the  meridian  at  the  same  instant  ;  so 
also  C  and  c,  D  and  d,  and  so  on. 

Suppose  now  that  a  fictitious  sun  moves  in  the  equator  at  the 
rate  of  59'  8"  per  day,  while  the  real  sun  moves  in  the  ecliptic  at 
the  same  rate,  and  let  them  start  together  from  A  at  noon  on  the 
20th  of  March.  On  the  21st  of  March,  at  noon,  the  real  sun  will 
have  advanced  toward  B  59'  8",  which  distance  projected  on  the 
equator  will  be  less  than  59'  8",  while  the  fictitious  sun  will  have 
advanced  toward  I  59'  8";  that  is,  the  fictitious  sun  will  be  east- 


EQUATION   OF  TIME.  71 

•ward  of  the  real  sun,  and  the  real  sun  will  come  to  the  meridian 
sooner  than  the  fictitious  one.  The  same  will  happen  during  the 
motion  of  the  sun  through  the  entire  quadrant  AG.  The  two 
suns  will  reach  the  points  G  and  g  on  the  21st  of  June,  and  then 
they  will  both  come  to  the  meridian  at  the  same  instant. 

During  the  motion  of  the  sun  through  the  second  quadrant,  the 
real  sun  will  come  to  the  meridian  later  than  the  fictitious  one,  but 
both  will  reach  the  point  N  on  the  22d  of  September  at  the  same 
instant.  During  the  motion  through  the  third  quadrant,  the  real 
sun  will  come  to  the  meridian  sooner  than  the  fictitious  one,  until 
the  21st  of  December,  when  they  will  be  found  180°  from  the 
points  G  and  g.  During  the  motion  through  the  last  quadrant, 
the  real  sun  will  come  to  the  meridian  later  than  the  fictitious 
one,  but  both  will  reach  the  point  A  at  the  same  instant  on  the 
21st  of  March.  Thus  we  see  that,  so  far  as  it  depends  upon  the 
obliquity  of  the  ecliptic,  the  equation  of  time  is  positive  for  three 
months ;  then  negative  for  three  months ;  then  positive  for  three 
months ;  and  then  negative  for  another  three  months. 

The  amount  of  the  equation  of  time  due  to  this  cause,  may  be 
computed  as  follows :  Suppose  the  sun  to  have  advanced  45°  from 
A ;  then,  in  the  right-angled  triangle  ADc?,  the  angle  at  A  is  23° 
27',  and  the  hypothenuse  is  45°.  Ac?  is  then  computed  from  the 
equation,  tang.  Ac?=cos.  A  tang.  AD, 

whence  Ad  is  found  to  be  42°  31'  47". 

The  difference  between  AD  and  Ad  is  2°  28'  13",  or  9m.  52.8s. 
in  time ;  and  this  is  about  the  greatest  amount  of  the  equation  of 
time,  due  to  the  obliquity  of  the  ecliptic. 

123.  Resulting  values  of  the  equation  of  time. — The  influence  of 
each  of  these  causes  upon  the  equation  of  time,  is  artificially  rep- 
resented in  the  following  figure,  where  AE  is  supposed  to  repre- 
sent a  year  divided  into  twelve  equal  parts  to  represent  the 
months ;  and  the  ordinates  of  the  curve  ABODE,  measured  from 
the  line  AE  as  an  axis,  represent  the  values  of  the  equation  of 
time,  so  far  as  it  depends  upon  the  unequal  motion  of  the  earth 
in  its  orbit;  and  the  ordinates  of  the  curve  FGHI  represent  the 
values  of  the  equation  of  time,  so  far  as  it  depends  upon  the  in- 
clination of  the  equator  to  the  ecliptic.  The  actual  equation  of 
time  will  be  found  by  taking  the  algebraic  sum  of  the  effects  due 
to  these  two  separate  causes.  The  result  is  the  curve  MNOPQB^ 


ASTRONOMY. 
Fig.  34. 


Jan,. 

Fib. 

Wardi. 

April. 

J% 

June. 

JWj: 

Aug. 

^r 

^ 

jsr.* 

/>«. 

£ 

y 

B 

^^ 

x^ 

C 

fl^ 

N 

] 

^ 

\ 

"~~"^ 

/ 

%* 

x 

XL 

r^ 

^X 

X 

i—  ^ 

. 

D 

1 

"Ss^1 

o 

\ 

/ 

[/ 

the  ordinates  being  measured  from  the  lower  horizontal  line  in 
the  figure.  From  this  we  see  that  the  equation  of  time  has  two 
annual  maxima  and  two  annual  minima,  and  there  are  four  pe- 
riods when  the  equation  is  zero.  These  dates  and  the  correspond- 
ing values  of  the  equation  of  time  are  as  follows : 


July  26,  -f   6m. 

September  1,  0 
November  2,  —16 
December  24,  0 


12s. 

0 
18 

0 


February  11,  +14m.  32s. 

April  15,  0        0 

May  14,  -  3      55 

June  14,  0        0 

These  dates  and  the  values  of  the  equation  of  time  change 
slightly  from  one  year  to  another,  so  that,  where  great  accuracy  is 
required,  a  table  of  the  equation  of  time  is  required  for  each  year. 
Such  a  table  is  annually  published  in  the  Nautical  Almanac. 

The  Calendar. 

124.  The  Julian  Calendar. — The  interval  between  two  success- 
ive returns  of  the  sun  to  the  vernal  equinox,  is  called  a  tropical 
year.  Its  average  length  expressed  in  mean  solar  time  is  365d. 
5h.  48m.  47.8s.  But  in  reckoning  time  for  the  common  purposes 
of  life,  it  is  most  convenient  to  have  the  year  contain  a  certain 
number  of  whole  days.  In  the  calendar  established  by  Julius  Ca3- 
sar,  and  hence  called  the  Julian  Calendar,  three  successive  years 
were  made  to  consist  of  365  days  each,  and  the  fourth  of  366 
days.  The  year  which  contained  366  days  was  called  a  bissextile 
year,  because  the  6th  of  the  Kalends  of  March  was  twice  counted. 
It  is  also  frequently  called  leap-year.  The  others  are  called  com- 
mon years.  The  odd  day  inserted  in  a  bissextile  year  is  called 
the  intercalary  day. 

The  reckoning  by  the  Julian  calendar  supposes  the  length  of 


THE   CALENDAR.  73 

the  year  to  be  365J  days.  A  Julian  year,  therefore,  exceeds  the 
tropical  year  by  llm.  12s.  This  difference  amounts  to  a  little 
more  than  3  days  in  the  course  of  400  years. 

125.  The  Gregorian  Calendar. — At  the  time  of  the  Council  of 
Nice,  in  the  year  325,  the  Julian  calendar  was  introduced  into  the 
Church,  and  at  that  time  the  vernal  equinox  fell  on  the  21st  of 
March ;  but  in  the  year  1582  the  error  of  the  Julian  calendar  had 
accumulated  to  nearly  10  days,  and  the  vernal  equinox  fell  on  the 
llth  of  March.    If  this  erroneous  reckoning  had  continued,  in  the 
course  of  time  spring  would  have  commenced  in  September,  and 
summer  in  December.     It  was  therefore  resolved  to  reform  the 
calendar,  which  was  done  by  Pope  Gregory  XIII.,  and  the  first 
step  was  to  correct  the  loss  of  the  ten  days,  by  counting  the  day 
after  the  4th  of  October,  1582,  not  the  5th,  but  the  15th  of  the 
month.    In  order  to  keep  the  vernal  equinox  to  the  21st  of  March 
in  future,  it  was  concluded  that  three  intercalary  days  should  be 
omitted  every  four  hundred  years.     It  was  also  agreed  that  the 
omission  of  the  intercalary  days  should  take  place  in  those  years 
which  were  not  divisible  by  400.    Thus  the  years  1700, 1800,  and 
1900,  which,  according  to  the  Julian  calendar,  would  be  bissextile, 
would,  according  to  the  reformed  calendar,  be  common  years. 

The  calendar  thus  reformed  is  called  the  Gregorian  Calendar. 
The  error  of  this  calendar  amounts  to  less  than  one  day  in  3000 
years. 

126.  Adoption  of  the  Gregorian  Calendar. — The  Gregorian  calen- 
dar was  immediately  adopted  at  Eome,  and  soon  afterward  in  all 
Catholic  countries.     In  Protestant  countries  the  reform  was  not 
so  readily  adopted,  and  in  England  and  her  colonies  it  was  not 
introduced  till  the  year  1752.    At  this  time  there  was  a  difference 
of  11  days  between  the  Julian  and  Gregorian  calendars,  in  conse- 
quence of  the  suppression  in  the  latter,  of  the  intercalary  day  in 
1700.    It  was  therefore  enacted  by  Parliament  that  11  days  should 
be  left  out  of  the  month  of  September  in  the  year  1752,  by  calling 
the  day  following  the  2d  of  the  month,  the  14th  instead  of  the  3d. 

The  Gregorian  calendar  is  now  used  in  all  Christian  countries 
except  Russia.  The  Julian  and  Gregorian  calendars  are  frequent- 
ly designated  by  the  terms  old  style  and  new  style.  In  consequence 
of  the  intercalary  days  omitted  in  the  years  1700  and  1800,  there 
is  now  12  davs  difference  between  the  two  calendars. 


74  ASTRONOMY. 

127.  When  does  the  year  begin? — In  the  different  countries  of 
Europe,  the  year  has  not  always  been  regarded  as  commencing  at 
the  same  date.     In  certain  countries,  the  year  has  been  regarded 
as  commencing  at  Christmas,  on  the  25th  of  December ;  in  others, 
on  the  1st  of  January ;  in  others,  on  the  1st  of  March ;  in  others, 
on  the  25th  of  March ;  and  in  others  at  Easter,  which  may  corre- 
spond to  any  date  between  March  22d  and  April  25th.     In  En- 
gland, previous  to  the  year  1752,  the  legal  year  commenced  on 
the  25th  of  March ;  but  the  same  act  that  introduced  the  Grego- 
rian calendar  established  the  1st  of  January  as  the  commencement 
of  the  year.    In  this  manner  the  year  1751  lost  its  month  of  Jan- 
uary, its  month  of  February,  and  the  first  24  days  of  March.    This 
change  in  the  calendar  explains  the  double  date  which  is  frequent- 
ly found  in  English  books.     For  example,  Feb.  15,  Trli>  means 
the  15th  of  February,  1751,  according  to  the  old  mode  of  counting 
the  years  from  the  25th  of  March,  and  1752  according  to  the  new 
method  prescribed  by  Parliament.     In  order  to  distinguish  the 
one  mode  of  reckoning  from  the  other,  it  was  for  a  long  time  cus- 
tomary to  attach  to  each  date  the  letters  O.  S.  for  old  style,  or 
N.  S.  for  new  style.    Thus  the  date  of  General  Washington's  birth 
was  either  written  Feb.  11, 1731,  O.  S.,  or  Feb.  22, 1732,  K  S. 

128.  First  and  last  days  of  the  year. — Since  a  common  year  con- 
sists of  365  days,  or  52  weeks  and  1  day,  the  last  day  of  each  com- 
mon year  must  fall  on  the  same  day  of  the  week  as  the  first ;  that 
is,  if  the  year  begins  on  Sunday  it  will  end  on  Sunday.     But  if 
leap-year  begins  on  Sunday  it  will  end  on  Monday,  and  the  fol- 
lowing year  will  begin  on  Tuesday. 

PROBLEMS  ON  THE  CELESTIAL  GLOBE. 

129.  To  find  the  right  ascension  and  declination  of  a  star. 

Bring  the  star  to  the  brass  meridian ;  the  degree  of  the  merid- 
ian over  the  star  will  be  its  declination,  and  the  degree  of  the 
equinoctial  under  the  meridian  will  be  its  right  ascension.  Eight 
ascension  is  sometimes  expressed  in  hours  and  minutes  of  time, 
and  sometimes  in  degrees  and  minutes  of  arc. 

Ex.  Required  the  right  ascension  and  declination  of  Arcturus. 

130.  The  right  ascension  and  declination  of  a  star  being  given,  to 
find  the  star  upon  the  globe. 


PROBLEMS.  75 

Bring  the  degree  of  the  equator  which  marks  the  right  ascen- 
sion to  the  brass  meridian;  then  under  the  given  declination 
marked  on  the  meridian  will  be  the  star  required. 

Ex.  Eequired  the  star  whose  right  ascension  is  lOh.  1m.  7s.,  and 
declination  12°  37'  K 

131.  To  set  the  celestial  globe  in  a  position  similar  to  that  of  the 
heavens,  at  a  given  place,  at  a  given  day  and  hour. 

Set  the  brass  meridian  to  coincide  with  the  meridian  of  the 
place ;  elevate  the  pole  to  the  latitude  of  the  place ;  bring  the 
sun's  place  in  the  ecliptic  to  the  meridian,  and  set  the  hour  index 
at  12 ;  then  turn  the  globe  westward  until  the  index  points  to 
the  given  hour.  The  constellations  would  then  have  the  same 
appearance  to  an  eye  situated  at  the  centre  of  the  globe,  as  they 
have  at  that  moment  in  the  heavens. 

Ex.  Required  the  appearance  of  the  heavens  at  New  Haven, 
Lat.  41°  18',  June  20th,  at  10  o'clock  P.M. 

132.  To  determine  the  time  of  rising,  setting,  and  culmination  of  a 
star  for  any  given  day  and  place. 

Elevate  the  pole  to  the  latitude  of  the  place ;  bring  the  sun's 
place  in  the  ecliptic  for  the  given  day  to  the  meridian,  and  set  the 
hour  index  to  12.  Turn  the  globe  until  the  star  comes  to  the 
eastern  horizon,  and  the  hour  shown  by  the  index  will  be  the 
time  of  the  star's  rising.  Bring  the  star  to  the  brass  meridian, 
and  the  index  will  show  the  time  of  the  star's  culmination.  Turn 
the  globe  until  the  star  comes  to  the  western  horizon,  and  the  in- 
dex will  show  the  time  of  the  star's  setting. 

Ex.  Required  the  time  when  Aldebaran  rises,  culminates,  and 
sets  at  Cincinnati,  October  10th. 

133.  To  determine  the  position  oftheplaneis  in  the  heavens  at  any 
given  time  and  place. 

Find  the  right  ascension  and  declination  of  the  planets  for  the 
given  day  from  the  Nautical  Almanac,  and  mark  their  places 
upon  the  globe;  then  adjust  the  globe  as  in  Art.  131,  and  the  po- 
sition of  the  planets  upon  the  globe  will  correspond  to  their  posi- 
tion in  the  heavens.  We  may  then  determine  the  time  of  their 
rising  and  setting  as  in  Art.  132.  The  time  of  rising  and  setting 
of  a  comet  mav  be  determined  in  the  same  manner. 


76 


ASTRONOMY. 


CHAPTER  V. 

PARALLAX.— ASTRONOMICAL   PROBLEMS. 

134.  Diurnal  parallax  defined.— The  direction  in  which  a  celes- 
tial body  would  be  seen  if  viewed  from  the  centre  of  the  earth,  is 
called  its  true  place ;  and  the  direction  in  which  it  is  seen  from 
any  point  on  the  surface,  is  called  its  apparent  place.  The  arc  of 
the  heavens  intercepted  between  the  true  and  apparent  places— 
that  is,  the  apparent  displacement  which  would  be  produced  by 
the  transfer  of  the  observer  from  the  centre  to  the  surface,  is  called 
the  diurnal' parallax. 

Let  C  denote  the  centre  of  the 
earth  ;  P  the  place  of  the  observer 
on  its  surface;  M  an  object  seen  in 
the  zenith  at  P ;  M'  the  same  ob- 
ject seen  at  the  zenith  distance 
MPM';  and  M"  the  same  object 
seen  in  the  horizon. 

It  is  evident  that  M  will  appear 
in  the  same  direction  whether  it  be 
viewed  from  P  or  C.    Hence,  in  the 
zenith,  there  is  no  diurnal  parallax, 
and  there  the  apparent  place  of  an  object  is  its  true  place. 

If  the  object  be  at  M',  its  apparent  direction  is  PM',  while  its 
true  direction  is  CM',  and  the  parallax  corresponding  to  the  zenith 
distance  MPM'  will  be  PM'C. 

As  the  object  is  more  remote  from  the  zenith,  the  parallax  in- 
creases; apd  when  the  object  is  in  the  horizon,  as  at  M",  the  di- 
urnal parallax  becomes  greatest,  and  is  called  the  horizontal  paral- 
lax. It  is  the  angle  PM"C  which  the  radius  of  the  earth  subtends 
at  the  object 

It  is  evident  that  parallax  increases  the  zenith  distance,  and  con- 
sequently diminishes  the  altitude.  Hence,  to  obtain  the  true  ze- 
nith distance  from  the  apparent,  the  parallax  must  be  subtracted; 
and  to  obtain  the  true  altitude  from  the  apparent,  the  parallax 


PARALLAX.  77 

must  be  added.  The  azimuth  of  a  heavenly  body  is  not  affected 
by  parallax. 

.  135.  To  deduce  the  parallax  at  any  altitude  from  the  horizontal 
parallax. — In  the  triangle  CPM'  we  have 

CM' :  CP : :  sin.  CPM/(=sin.  MPM') :  sin.  GMT.       (1) 
Also,  in  the  triangle  CPM",  we  have 

CM":CP::.l:sin.CM"P.  (2) 

Hence  1 :  sin.  CM'T : :  sin.  MPM'(:  sin.  GMT, 

or  sin.  CMT=sin.  CM"P  x  sin.  MPM' ; 

that  is,  the  sine  of  the  parallax  at  any  altitude,  is  equal  to  the  product 
of  the  sine  of  tfie  horizontal  parallax,  by  the  sine  of  the  apparent  zenith 
distance. 

The  parallax  of  the  sun  and  planets  is  so  small  that  we  may, 
without  sensible  error,  employ  the  parallax  itself  instead  of  its 
sine ;  that  is,  the  parallax  at  any  altitude  is  equal  to  the  product  of 
the  horizontal  parallax,  by  the  sine  of  the  apparent  zenith  distance. 

136.  Relation  of  the  parallax  of  a  heavenly  body  to  its  distance. 
Let  us  put  z= the  zenith  distance  MPM' ;        • 

p=ihe  parallax  CM'P; 
r=CP,  the  radius  of  the  earth  ; 
E^CM',  the  distance  of  the  heavenly  body. 
Then,  by  equation  (1), 

R  :  r ::  sin.  z  :  sin.^ 

or  sin.£>=j5-  sin.z, 

r  •  . 
or  P=-v  sin*  z>  veiT  nearty' 

The  parallax  at  any  given  altitude  varies,  therefore,  inversely 
as  the  distance,  very  nearly. 

When  the  zenith  distance  becomes  90°,  sin.  z  becomes  unity 
and  if  we  denote  the  horizontal  parallax  by  P,  we  shall  have 

sin.P=±, 

or  P=4»  very  nearly. 

±i 

137.  To  determine  the  parallax  of  the  moon  by  observation. — Let 
A,  A'  be  two  places  on  the  earth  situated  under  the  same  merid- 


78  ASTRONOMY. 

^  z          rig.  ST.  ian>  and  at  a  great  distance  from 

each  other;  let  C  be  the  centre  of 
the  earth,  and  M  the  moon. 

Let  AC  be  denoted  by  r,  and  CM 
by  E,  and  let  ZAM,  Z'A'M,  which 
are  the  moon's  zenith  distances  as 
measured  at  the  two  observatories, 
be  denoted  by  2,  and  z'.    Then  the 
moon's  parallax,  AMC,  at  the  station  A,  will  be 
r    . 
E 
and  the  parallax  A'MC  at  the  station  A'  will  be 


Adding  these  equations  together,  we  find 
'  =     (sin.  2+  sin. 


But  the  angle  p+p',  or  AMA',  is  equal  to  the  difference  be- 
tween ZCZ'  and  the  sum  of  the  angles  z  and  z'  ;  and  since,  if  the 
places  be  situated  one  north  and  the  other  south  of  the  equator, 
we  have  ZCZ'  equal  to  the  sum  of  the  latitudes  of  the  stations 
Z+Z',  we  obtain  p+p'=z+z'—l—l'. 

Substituting  this  value  in  the  preceding  equation,  we  find 

z+z'—  I—  Z'  =  T>  (sin.  z  +  sin.  z'}, 

-1-1' 


,. 

E    sm.z+sm.z 

But  =7  is  the  horizontal  parallax  of  the  moon,  which  was  re- 
quired to  be  found. 

138.  Stations  of  observation.  —  It  is  not  essential  that  the  two  ob- 
servers should  be  exactly  on  the  same  meridian  ;  for  if  the  me- 
ridian zenith  distances  of  the  moon  be  observed  on  several  consec- 
utive days,  its  change  of  meridian  zenith  distance  in  a  given  time 
will  be  known.  Then,  if  the  difference  of  longitude  of  the  two 
places  is  known,  the  zenith  distance  of  the  moon  as  observed  at 
one  of  the  meridians,  may  be  reduced  to  what  it  would  have  been 
found  to  be,  if  the  observations  had  been  made  in  the  same  lati- 
tude at  the  other  meridian. 


PARALLAX.  79 

139.  Results  obtained  ly  this  method. — There  is  an  observatory 
at  the  Cape  of  Good  Hope,  in  Lat.  33°  56'  S.,  where  the  moon's 
meridian  altitude  has  been  observed  daily  for  many  years,  when- 
ever the  weather  would  permit ;  and  similar  observations  are  reg- 
ularly made  at  Greenwich  Observatory,  in  Lat.  51°  28'  N".,  as  also 
at  numerous  other  observatories  in  Europe.     By  combining  these 
observations,  the  moon's  parallax  has  been  ascertained  with  great 
precision.     It  is  found  that  the  parallax  varies  considerably  from 
one  day  to  another.     The  equatorial  parallax,  when  greatest,  is 
about  61'  32",  and  when  least,  53'  48".     Its  average  value  is 
57'  2". 

By  the  preceding  method  the  sun's  parallax  may  be  ascertained 
to  be  about  9".  It  can,  however,  be  found  more  accurately  by  ob- 
servations of  the  transits  of  Venus,  as  will  be  explained  hereafter. 

The  parallax  of  the  planets  can  also  be  determined  in  the  same 
manner  as  that  of  the  moon ;  but  in  the  case  of  the  nearest  planet 
the  parallax  never  exceeds  32",  and  that  of  the  remoter  planets 
never  amounts  to  1" ;  and  there  are  other  methods  by  which  these 
quantities  can  be  more  accurately  determined. 

140.  To  compute  the  distance  of  a  heavenly  body. — When  we  know 
the  earth's  radius  and  the  horizontal  parallax  of  a  heavenly  body, 
we  can  compute  its  distance.     For  (Fig.  36) 

sin.  PM"C  :  PC  : :  radius  :  CM", 

or  the  distance  of  the  object  equals  the  radius  of  the  earth,  divided 
by  the  sine  of  the  horizontal  parallax. 

141.  Effect  of  the  ellipticity  of  the  earth  upon  parallax. — The  hor- 
izontal parallax  of  the  moon  is  the  angle  which  the  earth's  radius 
would  subtend  to  an  observer  at  the  moon.     On  account  of  the 
spheroidal  figure  of  the  earth,  this  horizontal  parallax  is  not  the 
same  for  all  places  on  the  earth,  but  varies  with  the  earth's  radius, 
being  greatest  at  the  equator,  and  diminishing  as  we  proceed  to- 
ward either  pole.     It  is  necessary,  therefore,  always  to  compute 
the  earth's  radius  for  the  place  of  the  observer,  and  this  may  be 
done  from  the  known  properties  of  an  ellipse.     The  moon's  hori- 
zontal parallax  for  any  given  latitude  is  equal  to  the  horizontal 
parallax  at  the  equator  multiplied  by  the  radius  of  the  earth  at 
the  given  latitude,  the  radius  ofHhe  equator  being  considered  as 
unity. 


80  ASTRONOMY. 

It  is  this  corrected  value  of  the  equatorial  parallax  which 
should  be  employed  in  all  computations  which  involve  the  par- 
allax of  a  particular  place. 

ASTRONOMICAL   PROBLEMS. 

142.  To  find  the  latitude  of  any  place. — The  latitude  of  a  place 
may  be  determined  by  measuring  the  altitude  of  any  circumpolar 
star,  both  at  its  upper  and  lower  culminations,  as  explained  in 
Art  76.  It  may  also  be  determined  by  measuring  a  single  me- 
ridian altitude  of  any  celestial  body  whose  declination  is  known. 
Fi,  33.  Let  S  or  S'  be  a  star  on  the  meridian  ; 

SE  or  S'E  its  declination.  Measure  SII, 
the  altitude  of  the  star  S,  and  correct  it 
for  refraction.  Then 


H  o  °       But  EH  is  the  complement  of  PO, 

which  is  the  latitude  sought.  The  declinations  of  all  the  brighter 
stars  have  been  determined  with  great  accuracy,  and  are  recorded 
in  catalogues  of  the  stars. 

143.  To  find  the  latitude  at  sea. — At  sea  the  latitude  is  usually 
determined  by  observing  with  the  sextant  the  greatest  altitude 
of  the  sun's  lower  limb  above  the  sea  horizon  at  noon.  The  ob- 
servations are  commenced  about  half  an  hour  before  noon,  and 
the  altitude  of  the  sun  is  repeatedly  measured  until  the  altitude 
ceases  to  increase.  This  greatest  altitude  is  considered  to  be  the 
altitude  on  the  meridian.  To  this  altitude  we  must  add  the  sun's 
semi-diameter  in  order  to  obtain  the  altitude  of  the  sun's  centre, 
and  this  result  must  be  corrected  for  refraction.  To  this  result  we 
must  add  the  sun's  declination  if  south  of  the  equator,  or  subtract 
it  if  north,  and  we  shall  obtain  the  elevation  of  the  equator, 
which  is  the  complement  of  the  latitude.  The  Nautical  Almanac 
furnishes  the  sun's  declination  for  every  day  of  the  year. 

'144.  To  find  the  time  at  any  place. — The  time  of  apparent  noon 
is  the  time  of  the  sun's  meridian  passage,  and  is  most  convenient- 
ly found  by  means  of  a  transit  instrument  adjusted  to  the  merid- 
ian. Mean  time  may  be  derived  from  apparent  time  by  applying 
the  equation  of  time  with  its  proper  sign. 

The  time  of  apparent  noon  may  also  be  found  by  noting  the 


ASTRONOMICAL  PROBLEMS.  81 

times  when  the  sun  has  equal  altitudes  before  and  after  passing 
the  meridian,  and  bisecting  the  interval  between  them.  When 
great  accuracy  is  required,  the  result  obtained  by  this  method  re- 
quires a  slight  correction,  since  the  sun's  declination  changes  be- 
tween morning  and  evening. 

145.  To  find  the  time  by  a  single  altitude  of  the  sun. — The  time 
may  also  be  computed  from  an  altitude  of  the  sun  measured  at 
any  hour  of  the  day,  provided  we  know  the  sun's  declination  and 
the  latitude  of  the  place. 

Let  PZH  be  the  meridian  of  the  place 

Fig.  30. 

of  observation,  P  the  pole,  Z  the  zenith, 
and  S  the  place  of  the  sun.  Measure  the 
zenith  distance,  ZS,  and  correct  it  for  re- 
fraction. Then,  in  the  spherical  triangle 
ZPS,  we  know  the  three  sides,  viz.,  PZ,  H 
the  complement  of  the  latitude,  PS,  the  distance  of  the  sun  from 
the  north  pole,  and  ZS,  the  sun's  zenith  distance.  In  this  triangle 
we  can  compute  (Trigonometry,  Art.  223)  the  angle  ZPS,  which, 
if  expressed  in  time,  will  be  the  interval  between  the  moment  of 
observation  and  noon.  This  observation  can  be  made  at  sea  with 
a  sextant,  and  this  is  the  method  of  determining  time  which  is 
commonly  practiced  by  navigators. 

146.  A  meridian  mark,  and  sun-dial — If,  upon  a  horizontal 
plane,  we  trace  a  meridian  line,  and  at  the  south  extremity  of 
this  line  erect  a  vertical  rod  freely  exposed  to  the  sun,  we  may 
determine  the  time  of  apparent  noon  by  the  passage  of  the  shadow 
of  the  rod  over  the  meridian  line.     Or,  if  we  set  up  a  straight  rod 
in  a  position  parallel  to  the  axis  of  the  earth,  its  shadow,  as  cast 
upon  a  horizontal  plane,  will  have  the  same  direction  at  any  given 
hour,  at  all  seasons  of  the  year.     If,  then,  we  graduate  this  hori- 
zontal plane  in  a  suitable  manner,  and  mark  the  lines  with  the 
hours  of  the  day,  we  may  determine  the  apparent  time  whenever 
the  sun  shines  upon  the  rod.     Such  an  instrument  is  called  a  sun- 
dial, and  it  may  be  constructed  with  sufficient  precision  to  answer 
the  ordinary  purposes  of  society.     This  instrument  will  always 
indicate  apparent  time ;  but  mean  time  may  be  deduced  from  it 
by  applying  the  equation  of  time. 

F 


82 


ASTRONOMY. 


Fig.  40. 


147.  To  compute  the  longitude,  right  ascension,  and  declination  of 
the  sun,  any  one  of  these  quantities,  together  with  the  obliquity  of  the 

ecliptic,  being  given. 

Let  EPQP'  represent  the  equinoc- 
tial colure,  EMQ  the  equator,  ESQ 
the  ecliptic,  E  the  first  point  of  Aries, 
S  the  place  of  the  sun,  PSP'  an  hour 
circle  passing  through  the  sun ;  then 
\<l  EM  is  the  sun's  right  ascension,  SM 
his  declination,  ES  his  longitude,  and 
MES  the  obliquity  of  the  ecliptic. 
Then,  in  the  triangle  ESM,  we  have, 
by  Napier's  rule, 

E  cos.  E= tang.  ME  cot.  SE ; 
that  is,  representing  the  obliquity  by  to,  and  the  right  ascension 
by  E.  A.  tang.  E.  A.  =  tang.  Long.  cos.  to,  (1) 

tanjy.R.A.  /ox 

and  tang.Long.-— ^-— .  (2) 

Also,  E  sin.  ME = tang.  MS  cot.  E ; 

that  is,  sin.  R.  A.  =  tang.  Dec.  cot.  w,                           (3) 

and  tang.  Dec. = sin.  E.  A.  tang.  to.                         (4) 

Also,  Esin.MS=sin.Esin.ES; 

that  is,  sin.  Dec. = sin.  &  sin.  Long.,                           (5) 

sin.  Dec.    , 

and  •  sin.  Long.  =  —- : .  (6) 

si  n .  ( i » 

Also,  E  cos. ES= cos.  ME  cos. MS; 

that  is,  cos.  Long. = cos.  E.  A.  cos.  Dec.,  (7) 

cos.  Long. 

and  cos.E.A.=—        -5$.  (8) 

cos.  Dec. 

Ex.  1.  On  the  1st  of  June,  1864,  at  Greenwich  mean  noon,  the 
sun's  right  ascension  was  4h.  38m.  27.75s.,  and  his  declination 
'  55".2  N. ;  required  his  longitude.  Ans.  71°  10'  35".9. 
•.  2.  On  the  1st  of  January,  1864,  the  sun's  longitude  was 
280°  23'  52".3,  and  his  declination  23°  2'  52".2  S. ;  required  his 
light  ascension.  Ans.  18h.  45m.  14.70s. 

Ex.  3.  On  the  20th  of  May,  1864,  the  sun's  longitude  was 
59C  40'  1".6,  and  the  obliquity  of  the  ecliptic  23°  27'  18".5 ,  re- 
quired  his  right  ascension  and  declination. 

4ns.R.A.3h.49m.  52.62s. 
Dec.  20°  5'  33".9  N. 


ASTRONOMICAL   PROBLEMS.  83 

Ex.  4.  On  the  27th  of  October,  1864,  the  sun's  right  ascension 
was  14h.  8m.  19.06s.,  and  the  obliquity  of  the  ecliptic  23°  27'  17".8; 
required  his  longitude  and  declination. 
-4*vu  $hA.*S^<&  &w«xX-vdJ  Ans.Long.  214°  20'  34".  7. 

:  X  «-*£<£  Dec-  12°  58'  34".4  S' 

Ex.  5.  On  the  8th  of  Augustfl864,  the  sun's  declination  was 
16°  0'  56".4  N.,  and  the  obliquity  of  the  ecliptic  23°  27'  18".2  ;  re- 
quired his  right  ascension  and  longitude. 

^*.  E.  A.  9h.  14m.  19.20s. 
Long.  136°  7'  6".5. 


-  148.  Given  the  latitude  of  a  place  and  the  surfs  declination,  to  find 
the  time  of  his  rising  or  setting. 

Let  PEP'  represent  the  hour  circle,  rig  41. 
which  is  six  hours  from  the  meridian, 
and  which  intersects  the  horizon  in  the 
east  point,  E.  Let  S  or  S'  be  the  posi- 
tion of  the  sun  in  the  horizon,  and 
through  S  draw  the  hour  circle  PSP'  ; 
also  through  S'  draw  the  hour  circle 
PS'P'.  Then,  in  the  right-angled  spher- 
ical triangle  EMS,  or  EM'S', 

EM  or  EM'  =  the  distance  of  the  sun  from  the  six  o'clock 

hour  circle. 
MS  or  M'S'=the  sun's  declination,  which  we  will  represent 

by& 

MES  =  M'ES/—  the  complement  of  the  latitude. 
Now,  by  Napier's  rule, 

E  sin.  EM  =  tang.  MS  cot.  MES. 
Representing  the  latitude  by  0, 

sin.  EM  =  tang.  S  tang.  0. 
The  time  from  the  sun's  rising  to  his  passing  the  meridian  =6 

hours  ±  EM. 

Ex.  1,  Required  the  time  of  sunrise  at  New  York,-Lat.  40°  42', 
on  the  10th  of  May,  when  the  sun's  declination  is  17°  49'  N. 

Ans.  4h.  56m. 

Ex.  2.  Required  the  time  of  sunset  at  Cincinnati,  Lat.  39°  6',  on 
the  5th  of  November,  when  the  sun's  declination  is  15°  56'  S. 

Ans.  oh.  6m.  apparent  time. 
° 


84  ASTRONOMY. 

149.  To  find  the  time  when  the  surfs  upper  limb  rises,  allowance 
being  made  for  refraction. — The  preceding  method  gives  the  time 
when  the  sun's  centre  would  rise  if  there  were  no  refraction.  The 
effect  of  refraction  is  to  cause  the  sun  to  be  seen  above  the  sensi- 
ble horizon  sooner  in  the  morning,  and  later  in  the  afternoon,  than 
he  actually  is ;  and  moreover,  when  the  sun's  upper  limb  coincides 
with  the  horizon,  the  centre  is  about  16'  below.  At  the  instant, 
therefore,  of  sunrise  or  sunset,  his  centre  is  90°  50'  from  the  zenith ; 
the  semi-diameter  being  about  16',  and  the  horizontal  refraction 
34'.  In  order,  therefore,  to  compute  the  apparent  time  of  rising 
of  the  sun's  upper  limb,  we  must  compute  when  the  sun's  centre  is 
90°  50'  from  the  zenith.  This  may  be  done  as  follows : 
Fig.  42.  z  Let  PZH  be  the  meridian  of  the  place  of 

observation,  P  the  pole,  Z  the  zenith,  and 
S  the  place  of  the  sun.  In  the  spherical 
triangle  ZPS,  the  three  sides  are  known, 

H o  viz., 

PZ^the  co-latitude =^; 
ZS=the  zenith  distance =2; 
PS=the  north  polar  distance  of  the  sun=c?. 
In  this  triangle  we  can  compute  ZPS,  which  is  the  angular  dis- 
tance of  the  sun  from  the  meridian. 
By  Trigonometry, 

.  (S— 1)}  sin.  (S— c) 


sin.  b  sin.  c 
Put 

then  sin.  •£?=•., 

sin.  -^  sin. 

Ex.  1.  Required  the  time  of  sunset  at  New  York,  Lat.  40°  42', 

on  the  10th  of  May,  when  the  sun's  declination  is  17°  49'  N. 

Here      ^=  49°  18'      .  sin.  (S-^)=9.922S92 

d=  72    11  sin.  (S-(/)=9.747281 

z  =  90    50  cosec.  $ = 0. 120254 

S=106      9£  cosec.  d= 0021345 

S— $=  56    51£  2)9.811772" 

S-<2=  33    58^          £P=  53°  37£'  sin.  =9^905886 

P=107°15'=7h.9m. 

Hence  the  sun  sets  at  7h.  9m.  apparent  time ;  or,  subtracting  4ra 
for  equation  of  time,  we  have  7h.  5m.  moan  time. 


ASTRONOMICAL  PROBLEMS.  85 

Ex.  2.  Required  the  mean  time  of  sunrise  at  Boston,  Lat  42°  21', 
on  the  loth  of  October,  when  the  sun's  declination  i&-8°  47'  S., 
mean  time  being  14  minutes  slow  of  apparent  time. 

Ans.  6h.  14m. 

150.  To  find  the  time  of  beginning  or  end  of  twilight. — At  the  be- 
ginning or  end  of  twilight,  the  sun  is  18°  below  the  horizon ;  that 
is,  his  zenith  distance  is  108°.     Hence  this  problem  can  be  solved 
by  the  formula  of  the  last  article. 

Ex.  1.  Eequired  the  time  of  the  commencement  of  twilight  at 
Washington,  Lat.  38°  53',  on  the  1st  of  June,  when  the  sun's  dec- 
lination is  22°  10'  N,  mean  time  being  2  minutes  slow  of  appa- 
rent time.  Ans.  2h.  41m.  mean  time. 

Ex.  2.  Required  the  time  of  ending  of  twilight  at  New  Orleans, 
Lat.  29°  57',  on 'the  19th  of  February,  when  the  sun's  declination 
is  11°  19'  S.,  mean  time  being  14  minutes  fast  of  apparent  time. 

Ans.  7h.  12m.  mean  time. 

151.  To  compute  the  distance  between  two  stars  whose  rigid  ascen- 
sions and  declinations  are  knoiun. 

Let  P  be  the  pole,  and  S  and  S'  two  stars  whose     Fig- 4a 
places  are  known.     Then  PS  and  PS'  will  repre- 
sent their  polar  distances,  and  SPS'  will  be  the 
difference  of  their  right  ascensions.     Draw  SM 
perpendicular  to  PS'  produced.     Then 

R  cos.  P=tang.  PM  cot.  PS. 
Therefore,    tang.  PM = cos.  P  tang.  PS. 

Also,  S'M=PM-PS'. 

And    cos.  PM  :  cos.  S'M  : :  cos.  PS  :  cos.  S'S. 

Ex.  1.  Required  the  distance  from  Aldebaran,  R.  A.  4h.  27m. 
25.9s.,  polar  distance  73°  47'  33",  to  Sirius,  R.  A.  6h.  38m.  37.6s., 
polar  distance  106°  31'  2". 

P  =  2h.  llm.  11.7s.  =  32°  47'  55"  cos.  =  9. 9245 79 
PS =106  31  2  tang. =0527916 
PM=109  25  55  tang.  =0.452495 
PS'=  73  47  33 

S'M=  35  38 "22  cos. =9.909930 
PS=106  31  2  cos.  =  9.453782 
PM  =  109  25  55  sec.  =  0.477964 
SS'=  46  0  44  cos.  =  9.841676 


86 


ASTRONOMY. 


Ex.  2.  Required  the  distance  from  Regulus,  R.  A.  lOb.  Om.  29.1s., 
polar  distance  77°  18'  41",  to  Antares,  R.  A.  16h.  20m.  20.3s.,  polar 

1100  K/  KZ."  A  <nc   QQ°  r».V  J..V 


distance  116°  5'  55 


Ans.  99 


Fig.  44. 


152.  Distance  between  two  stars  on  the  same  parallel  of  declination. 
— If  two  stars  have  the  same  declination,  their  distance  can  be 
computed  as  follows : 

Let  P  be  the  pole,  EQ  a  portion  of  the  equa- 
tor, and  SS'  a  portion  of  any  parallel  of  dec- 
lination, and  PCE,  PCQ  two  meridians  passing 
through  S  and  S'. 
Then,  by  Geometry, 

arc  EQ  :  arc  SS' : :  CQ  :  AS  : :  1 :  cos.  Dec. 
Therefore     SS' = EQ  cos.  Dec.  =  EPQ  cos.  Dec. 
That  is,  the  distance  between  the  two  stars  is  equal  to  their  dif- 
ference of  right  ascension,  multiplied  by  the  cosine  of  their  dec- 
lination.    This  distance  is,  however,  not  measured  on  an  arc  of  a 
great  circle,  but  on  a  parallel  of  declination. 


Fig.  45. 


153.  To  find  the  longitude  and  latitude  of  a  star,  when  its  right  as- 
cension and  declination  are  known. 

Let  P  represent  the  pole  of  the 
equator,  E  the  pole  of  the  ecliptic, 
C  the  first  point  of  Aries,  PSP'  an 
hour  circle  passing  through  the 
star  S,  and  ESE'  a  circle  of  lati- 
tude passing  through  the  same  star. 
Then  AEBE'  represents  the  solsti- 
tial colure,  EP  represents  the  ob- 
liquity of  the  ecliptic,  PS  the  polar 
distance  of  the  star,  ES  its  co-lati- 
tude; SPB  is  the  complement  of 
its  right  ascension,  and  SEB  is  the  complement  of  its  longitude. 
Draw  SM  perpendicular  to  PB.  Represent  PM  by  a ;  also  repre- 
sent the  longitude  of  the  star  S  by  L,  its  latitude  by  I,  and  the  ob- 
liquity of  the  ecliptic  by  w. 

Now,  by  Napier's  rule,  we  have 

R  cos.  SPM=tang.  PM  cot  PS ; 
that  is,  sin.  R.  A.  =tang.  a  tang.  Dec., 

or  tang.  a=sin.  R.  A.  cot  Dec.  (A) 


ASTRONOMICAL   PROBLEMS.  87 


Also,  EM 

Again,  Trig.,  Art.  216,  Cor.  3, 

sin.  EM  :  sin.  PM  :  :  tang.  SPM  :  tang.  SEM  ; 
that  is, 

sin.  (a+  w)  :  sin.  a  :  :  cot  R  A.  :  cot.  L  :  :  tang.  L  :  tang.  R  A., 

T     tansr.  R.  A.  xsin.  (a  +  w) 

or  tang.  L=  —  :  -  :  ---  v         '.  (1) 

sm.  a 

Also,  R  cos.  SEM  =  tang.  EM  cot.  ES  ; 

that  is,  tang.  ?=cot  (a  +  w)  sin.  L.  (2) 

Ex.  1.  On  the  1st  of  January,  1864,  the  R.  A.  of  Capella  was  5h. 
6m.  42.01s.,  and  its  Dec.  45°  51'  20".l  K  ;  required  its  latitude  and 
longitude,  the  obliquity  of  the  ecliptic  being  23°  27'  19".  45. 
By  equation  (A), 

RA.  76°  40'  30".15      sin.  =  9.988148 
Dec.  45    51    20  .1        cot  =  9.987028 
a  =43    21    48  .2      tang.  =9.975176 
(u  =  23    27    19  .45 


>  =  66    49      7  .65 
By  equation  (1), 


sin.  («  +  &,)  =  9.  963440 

cosec.  o  =  0.  163282 

L=79°5S'3".5     tang.  =0.752249 

By  equation  (2),  cot.  (a  +  w)  =  9.  63  1  659 

sin.  L=9.993308 

7=22°  51'  4S".3     tang.  =9.624967 

Ex.  2.  On  the  1st  of  January,  1864,  the  R  A.  of  Regulus  was 
lOh.  1m.  9.34s.,  and  its  Dec.  12°  37'  36".8  N.  ;  required  its  latitude 
and  longitude,  the  obliquity  of  the  ecliptic  being  23°  27'  19".  45. 

Ans.  Latitude, 
Longitude, 


ASTRONOMY. 


CHAPTER  VI. 

THE  SUN — ITS  PHYSICAL  CONSTITUTION. 

154  Distance  of  the  sun.  —  The  distance  of  the  sun  from  the 
earth  can  be  computed  when  we  know  its  horizontal  parallax,  and 
the  radius  of  the  earth. 

The  mean  value  of  the  horizontal  parallax  of  the  sun  has 
been  found  to  be  8".58,  and  the  equatorial  radius  of  the  earth  is 
3963  miles. 

Hence       sin.  8".58  :  3963  : :  1 :  the  sun's  distance, 
which  is  found  to  be  95,300,000  miles ;  or,  in  round  numbers,  95 
millions  of  miles. 

155.  Velocity  of  the  earth's  motion  in  its  orbit. — Since  the  earth 
makes  the  entire  circuit  around  the  sun  in  one  year,  its  daily  mo- 
tion may  be  found  by  dividing  the  circumference  of  its  orbit  by 
365£,  and  thence  we  may  find  the  motion  for  one  hour,  minute, 
or  second.     The  circumference  of  the  earth's  orbit  is  very  nearly 
that  of  a  circle  whose  radius  is  the  sun's  mean  distance.     We 
thus  find  the  circumference  of  the  orbit  to  be  598,800,000  miles; 
that  the  earth  moves  1,639,000  miles  per  day ;  68,300  miles  per 
hour;  1138  miles  per  minute;  and  nearly  19  miles  per  second. 

By  the  diurnal  rotation,  a  point  on  the  earth's  equator  is  car- 
ried round  at  the  rate  of  1037  miles  per  hour,  or  17  miles  per 
minute.  The  motion  in  the  orbit  is,  therefore,  66  times  as  rapid 
as  the  diurnal  motion  at  the  equator. 

156.  The  diameter  of  the  sun.— The  sun's  absolute  diameter  can 
be  computed,  when  we  know  his  distance  and  apparent  diameter. 
The  apparent  diameter,  as  well  as  the  distance,  is  variable,  but  the 
mean  value  of  his  apparent  diameter  is  32'  3".64.     Hence  we 
have  the  proportion 

rad. :  ES  (95  millions) : :  sin.  16'  1".8 :  sun's  radius, 
which  is  found  to  be  444,406  miles;  or  his  diameter  is  888,812 
miles. 

The  diameter  of  the  sun  is  therefore  112  times  that  of  the  earth; 


THE   SUN.  89 

Fig.  4C.  and,  since  spheres  are  as  the  cubes 

of  their  diameters,  the  volume  of 
the  sun  is  more  than  1,400,000 
times  that  of  the  earth. 

The  density  of  the  sun  is  about 
one  quarter  that  of  the  earth ;  and,  therefore,  his  mass,  which  is 
equal  to  the  product  of  his  volume  by  his  density,  is  found  to  be 
355,000  times  that  of  the  earth. 

157.  Figure  of  the  sun's  disc. — Since  the  sun  rotates  upon  an 
axis,  as  shown  Art.  169,  his  figure  can  not  be  that  of  a  perfect 
sphere.    The  oblateness  of  a  heavenly  body  depends  chiefly  upon 
the  ratio  of  the  centrifugal  force  to  the  force  of  gravity  upon  its 
surface.    Now,  on  account  of  its  slow  rotation,  the  centrifugal  force 
of  a  point  upon  the  sun's  equator,  is  only  about  one  sixth  what  it 
is  upon  the  earth,  while  the  force  of  gravity  is  nearly  thirty  times 
as  great ;  hence  the  oblateness  of  the  sun  should  be  only  about 
-j-l-yth  part  of  that  of  the  earth.    But  the  oblateness  of  the  earth  is 
about  Trijo-th.     Hence  the  oblateness  of  the  sun  should  be  only 
about  j-rinnj-j  which  corresponds  to  a  difference  of  less  than  one 
twentieth  part  of  a  second  between  the  equatorial  and  polar  diam- 
eters.    This  quantity  is  too  small  to  be  detected  by  our  observa- 
tions ;  and  although  the  sun's  diameter  has  been  measured  many 
thousand  times,  still,  with  the  exception  of  the  effect  due  to  re- 
fraction, explained  in  Art.  91,  his  disc  is  sensibly  a  perfect  circle. 

158.  Force  of  gravity  on  the  sun. — The  attraction  of  a  sphere  be- 
ing the  same  as  if  its  whole  mass  were  collected  in  its  centre,  will 
be  proportional  to  the  mass  directly,  and  the  square  of  the  dis- 
tance inversely ;  hence  the  force  of  gravity  on  the  surface  of  the 
sun,  will  be  to  the  force  of  gravity  on  the  surface  of  the  earth,  as 

SOT  000 

'  to  unity,  which  is  27.9  to  1 ;  that  is,  a  pound  of  terres- 
trial matter  at  the  sun's  surface,  would  exert  a  pressure  equal  to 
what  27.9  such  pounds  would  do  at  the  surface  of  the  earth.  A 
body  weighing  200  pounds  on  the  earth,  would  produce  a  press- 
ure of  5580  pounds  on  the  sun. 

At  the  surface  of  the  earth,  a  body  falls  through  16-nj-th  feet  in 
one  second ;  but  a  body  on  the  sun  would  fall  through  16-pj  x  27.9 
=448.7  feet  in  one  second. 


90  ASTRONOMY. 

PHYSICAL   CONSTITUTION   OF  THE   SUN. 

159.  Solar  spots. — When  we  examine  the  sun  with  a  good  tel- 
escope, we  frequently  perceive  upon  his  surface,  black  spots  of  ir- 
regular shape,  sometimes  extremely  minute,  and  at  other  times  of 
vast  extent    They  usually  make  their  first  appearance  at  the  east- 
ern limb  of  the  sun ;  advance  gradually  toward  the  centre ;  pass 
beyond  it,  and  disappear  at  the  western  limb,  after  an  interval  of 
about  14  days.     They  remain  invisible  about  14  days,  and  then 
sometimes  reappear  at  the  eastern  limb  in  nearly  the  same  posi- 
tion as  at  first,  and  again  cross  the  sun's  disc  as  before,  having 
taken  27d.  7h.  in  the  entire  revolution. 

The  appearance  of  a  solar  spot  is  that  of  an  intensely  black,  ir- 
regularly-shaped patch,  called  the  nucleus,  surrounded  by  a  fringe 
which  is  less  dark,  and  is  called  the  penumbra.  The  form  of  this 
fringe  is  generally  similar  to  that  of  the  inclosed  black  spot ;  but 
this  is  not  always  the  case,  for  several  dark  spots  are  occasionally 
included  in  a  common  penumbra. 

Black  spots  have  occasionally  been  seen  without  any  penum- 
bra ;  and  sometimes  we  see  a  large  penumbra  without  any  central 
nucleus ;  but  generally  both  the  nucleus  and  penumbra  are  com- 
bined. 

160.  Changes  of  the  spots. — These  spots  change  their  form  from 
day  to  day,  and  sometimes  from  hour  to  hour.     They  usually 
commence  from  a  point  of  insensible  magnitude,  grow  very  rapid- 
ly at  first,  and  sometimes  attain  their  full  size  in  less  than  a  day. 
Then  they  remain  stationary,  with  a  well-defined  penumbra,  and 
continue  for  ten,  twenty,  and  some  even  for  fifty  days.     Then  the 
nucleus  usually  becomes  divided  by  a  narrow  line  of  light;  this 
line  sends  out  numerous  branches,  which  extend  until  the  entire 
nucleus  is  covered  by  the  penumbra. 

Decided  changes  have  been  detected  in  the  appearance  of  a  spot 
within  the  interval  of  a  single  hour,  indicating  a  motion  upon 
the  sun's  surface  of  at  least  1000  miles  per  hour. 

The  duration  of  the  spots  is  very  variable.  A  spot  has  ap- 
peared and  vanished  in  less  than  24  hours,  while  others  have 
lasted  for  weeks,  and  even  months.  In  1840,  a  spot  was  identi- 
fied for  nine  revolutions,  which  corresponds  to  a  period  of  about 
eight  months. 


PHYSICAL   CONSTITUTION   OF  THE   SUN.  91 

'  161.  Afagnitude  and  number  of  the  spots.— Soldi'  spots  are  some- 
times  of  immense  magnitude,  so  that  they  have  repeatedly  been 
visible  to  the  naked  eye.  In  June,  18-13,  a  solar  spot  remained  for 
a  whole  week  visible  to  the  naked  eye.  Its  breadth  measured 
167",  which  indicates  an  absolute  diameter  of  77,000  miles. 

The  number  of  spots  seen  on  the  sun's  disc  is  very  variable. 
Sometimes  the  disc  is  entirely  free  from  them,  and  continues  thus 
for  weeks,  or  even  months  together;  at  other  times  a  large  por- 
tion of  the  sun's  disc  is  covered  with  spots.  Sometimes  the  spots 
are  small,  but  numerous ;  and  sometimes  they  appear  in  groups 
of  vast  extent.  In  a  large  group  of  spots  which  appeared  in  1846, 
upward  of  200  single  spots  and  points  were  counted.  In  1837  a 
cluster  of  spots  covered  an  area  of  nearly  5  square  minutes,  or 
nearly  4000  millions  of  square  miles. 

162.  The  black  nucleus. — It  is  not  certain  that  the  black  nucleus 
of  a  spot  is  entirely  destitute  of  light ;  for  the  most  intense  arti- 
ficial light,  when  seen  projected  on  the  sun's  disc,  appears  as  dark 
as  the  spots  themselves.     Sir  W.  Herschel  estimated  that  the  light 
of  the  penumbra  was  less  than  one  half  that  of  the  brighter  part 
of  the  sun's  surface,  and  the  light  of  the  nucleus  less  than  one 
hundredth  of  the  brighter  surface. 

163.  Upon  what  part  of  the  sun  do  the  spots  appear? — Spots  are 
generally  confined  to  a  zone  extending  about  30°  on  each  side  of 
the  sun's  equator,  and  there  are  only  three  cases  on  record  in 
which  spots  have  been  seen  as  far  as  45°  from  the  sun's  equator. 
They  are  rarely  seen  directly  upon  the  sun's  equator  or  nearer  to 
it  than  8°  of  latitude,  and  they  are  most  frequent  between  the 
parallels  of  8°  and  20°.     They  are  generally  more  numerous  and 
of  a  greater  size  in  the  sun's  northern  hemisphere  than  in  the 
southern  hemisphere.     When  groups  of  spots  are  very  numerous 
they  often  manifest  a  tendency  to  arrange  themselves  in  a  line 
nearly  parallel  to  the  sun's  equator,  and  they  sometimes  extend 
thus  almost  entirely  across  the  visible  disc.     Some  astronomers 
have  thought  there  was  a  particular  meridian  of  the  sun  near 
which  spots  are  unusually  frequent,  and  that  spots  are  generated 
again  and  again  at  distant  intervals  of  time  over  the  same  iden- 
tical points  of  the  sun's  surface ;  but  this  conjecture  has  not  been 
clearly  established. 


92 


ASTKONOMY. 


164 .  Appearance  of  the  bright  part  of  the  sun's  disc. — Independ- 
ently of  tbo  dark  spots,  the  luminous  part  of  the  sun's  disc  is  not 
uniformly  bright.     It  exhibits  a  mottled  appearance,  like  that 
which  would  be  presented  by  a  stratum  of  luminous  clouds  of  ir- 
regular shape  and  variable  depth.     This  mottled  appearance  is 
not  confined,  like  the  black  spots,  to  a  particular  zone,  but  is  seen 
on  all  parts  of  the  surface,  even  near  the  poles  of  rotation. 

Sometimes  we  observe  upon  the  sun's  disc  curved  lines,  or 
streaks  of  light,  more  luminous  than  the  rest  of  the  surface.  These 
are  called  faculoe,  and  they  generally  appear  in  the  neighborhood 
of  the  black  spots. 

165.  Proof  that  the  suris  outer  envelope  is  not  solid. — The  rapid 
changes  which  take  place  upon  the  surface  of  the  sun,  prove  that 
his  outer  envelope  is  not  solid.     Admitting  that  the  great  mass 
of  the  sun  is  solid,  that  portion  which  we  ordinarily  see,  must 
be  either  liquid  or  gaseous ;  and  the  rapid  motion  of  1000  miles 
per  hour,  which  has  been  observed  in  solar  spots,  indicates  that 
the  luminous  matter  which  envelops  the  sun  must  be  gaseous, 
since  liquid  bodies  could  hardly  be  supposed  to  move  with  such 
velocity. 

166.  The  solar  spots  are  not  planetary  bodies. — It  is  evident  that 
the  solar  spots  are  at  the  surface  of  the  sun;  for  if  they  were  bodies 
revolving  around  the  sun  at  some  distance  from  it,  the  time  dur- 
ing which  they  would  be  seen  on  the  sun's  disc 

would  be  less  than  that  occupied  in  the  remainder 
of  their  revolution.  Thus,  let  S  represent  the 
sun,  E  the  earth,  and  suppose  ABC  to  represent 
the  path  of  an  opaque  body  revolving  about  the 
sun.  Then  AB  represents  that  part  of  the  orbit 
in  which  the  body  would  appear  projected  upon 
the  sun's  disc,  and  this  is  less  than  half  the  entire 
circumference ;  whereas  the  spot  reappears  on  the 
opposite  limb  of  the  sun  after  an  interval  nearly 
equal  to  that  required  to  pass  across  the  disc. 

167.  The  dark  spots  are  depressions  in  the  lu- 
minous matter  which  envelops  the  sun.     This 
was  first  proved  by  an  observation  made  by  Dr. 


PHYSICAL  CONSTITUTION  OF  THE   SUN. 


93 


Wilson,  of  Glasgow,  in  November,  1769.  He  first  noticed  a  spot 
November  22d,  when  it  was  not  far  from  the  sun's  western  limb ; 
and  he  observed  that  the  penumbra  was  about  equally  broad  on 
every  side  of  the  nucleus.  The  next  day  the  eastern  portion  of 
the  penumbra  had  contracted  in  breadth,  while  the  other  parts  re- 
mained nearly  of  their  former  dimensions.  On  the  24th  the  pe- 


Fig.  43. 


numbra  had  entirely  disappeared  from  the  eastern  side,  while  it 
was  still  visible  on  the  western  side.  On  the  llth  of  December 
the  spot  reappeared  on  the  sun's  eastern  limb,  and  now  there  was 
no  penumbra  on  the  western  side  of  the  spot,  although  it  was  dis- 
tinctly seen  on  the  remaining  sides.  The  next  day  the  penum- 
bra came  into  view  on  the  western  side,  though  narrower  than  on 
the  other  sides.  On  the  17th  the  spot  had  passed  the  centre  of 
the  sun's  disc,  and  now  the  penumbra  appeared  of  equal  extent 
on  every  side  of  the  nucleus.  From  these  observations,  it  is  in- 
ferred that  the  penumbra  is  lower  than  the  general  level  of  the 
sun's  bright  surface,  and  the  nucleus  lower  than  the  penumbra. 
Dr.  Wilson  computed  that  the  depth  of  the  spot  just  described 
was  nearly  4000  miles. 


94  ASTRONOMY. 

Similar  observations  were  repeatedly  made  by  Sir  W.  Herschel. 
In  1794  he  observed  that,  as  a  spot  approached  near  the  western 
limb  of  the  sun,  the  black  nucleus  gradually  contracted  in  breadth, 
while  its  length  remained  unchanged.  It  became  reduced  to  a 
narrow  black  line,  and  then  disappeared,  while  the  penumbra  was 
still  visible.  Similar  observations  have  repeatedly  been  made  by 
other  astronomers. 

In  1801  Sir  W.  Herschel  observed  that  when  a  spot  came  near 
the  western  margin  of  the  sun,  he  was  able  to  distinguish  the 
thickness  of  the  stratum  on  the  western  border,  but  not  on  the 
eastern ;  and  he  hence  computed  that  the  depression  of  the  pe- 
numbra below  the  bright  surface  of  the  sun  was  not  less  than  1800 
miles.  Similar  observations  have  been  made  by  M.  Secchi  at 
Rome. 

168.  The  bright  streaks  orfaculce  are  elevated  ridges  rising  above 
the  general  level  of  the  sun's  surface.     This  is  proved  by  an  ob- 
servation made  in  1859  by  Mr.  Dawes,  of  England.     He  had  the 
good  fortune  to  observe  a  bright  streak  of  unusual  size  precisely 
at  the  edge  of  the  sun's  disc,  and  he  perceived  that  it  projected 
beyond  the  circular  outline  of  the  disc  in  the  manner  of  a  mount' 
ain  ridge. 

In  1862,  as  an  uncommonly  large  spot  was  passing  off  the  sun's 
disc,  Mr.  Hewlett  perceived  a  small  notch  in  the  sun's  margin, 
precisely  over  the  place  where  the  great  nucleus  had  previously 
been  seen,  and  on  either  side  of  it  the  photosphere  appeared  to  be 
heaped  up  above  the  general  level  of  the  sun's  surface. 

169.  To  determine  the  time  of  the  sun's  rotation. — It  is  found  that 

a  sp°t  generally  employs  27^  days  in 
passing  from  one  limb  of  the  sun  around 
to  the  same  limb  again,  and  it  is  inferred 
that  this  apparent  motion  is  caused  by  a 
rotation  of  the  sun  upon  his  axis.  But 
the  period  above  mentioned  is  not  the 
time  in  which  the  sun  performs  one  ro- 
tation about  his  axis  ;  for;  let  A  A'B  rep- 
resent the  sun,  and  EET>  the  orbit  of  the 
earth.  When  the  earth  is  at  E,  the  visi- 
ble disc  of  the  sun  is  A  A'B;  and  if  the  earth  was  stationary  at 


PHYSICAL  CONSTITUTION  OF  THE  SUN.  95' 

E,  then  the  time  required  for  a  spot  to  move  from  the  limb  B 
round  to  the  same  point  again  would  be  the  time  of  the  sun's 
rotation.  But  while  the  spot  has  been  performing  its  apparent 
revolution,  the  earth  has  advanced  in  her  orbit  from  E  to  E',  and 
now  the  visible  disc  of  the  sun  is  A'B',  so  that  the  spot  has  per- 
formed more  than  a  complete  revolution  in  the  time  it  has  taken 
to  move  from  the  western  limb  to  the  western  limb  again.  Since 
an  apparent  rotation  of  the  sun  takes  place  in  27^  days,  the  num- 

365 a 
ber  of  apparent  rotations  in  a  year  will  be        ?;  or  13.4. 

*'•* 

But*  in  consequence  of  the  motion  of  the  earth  about  the  sun, 
if  the  sun  had  no  real  rotation,  it  would  in  one  year  make  an  ap- 
parent rotation  in  a  direction  contrary  to  the  motion  of  the  earth. 
Hence,  in  one  year,  there  must  be  14.4  real  rotations  of  the  sun, 

and  the  time  of  one  real  rotation  is  -7;  HP  or  25-3  days.    Thus  the 

time  of  a  real  rotation  is  found  to  be  nearly  two  days  less  than 
that  of  an  apparent  rotation. 

170.  Temperature  of  different  parts  of  the  sun's  disc. — By  receiv- 
ing the  image  of  different  portions  of  the  sun  upon  a  very  sensi- 
tive thermometer,  it  has  been  discovered  that  the  sun's  disc  has 
not  throughout  exactly  the  same  temperature.     The  rays  pro- 
ceeding from  the  centre  of  the  disc  are  hotter  than  those  which 
proceed  from  the  margin,  and  the  black  spots  radiate  less  heat 
than  the  neighboring  bright  surface. 

The  luminous  intensity  of  different  portions  of  the  sun's  disc 
exhibits  corresponding  variations,  the  borders  of  the  disc  being 
found  less  luminous  than  the  centre.  This  difference  is  quite 
noticeable  in  a  photographic  picture  of  the  sun. 

171.  Influence  of  solar  spots  upon  terrestrial  temper  attires. — It  has 
been  supposed  that  the  presence  of  an  unusual  number  of  large 
spots  on  the  sun's  disc  must  influence  the  temperature  of  the 
earth,  and  there  are  some  facts  which  favor  this  supposition.     At 
Paris,  out  of  26  years  of  observations,  the  mean  temperature  of 
those  years  in  which  the  spots  were  most  numerous  was  half  a 
degree  lower  than  that  of  those  years  in  which  the  spots  were 
least  frequent.     But  during  the  same  years  a  slight  effect  of  the 
opposite  kind  was  observed  upon  the  temperature  of  places  in  the 


96  ASTRONOMY. 

United  States,  so  that  we  seem  obliged  to  ascribe  the  differences 
in  question  to  other  causes  than  the  solar  spots. 

172.  Position  of  Hie  sun's  equator. — Besides  the  time  of  rotation, 
observations  of  the  solar  spots  enable  us  to  ascertain  the  position 
of  the  equator  with  reference  to  the  ecliptic.    The  angle  between 
the  solar  equator  and  the  ecliptic  has  been  determined  to  be 
about  7°.     About  the  first  weeks  of  June  and  December,  the 
spots,  in  traversing  the  sun's  disc,  appear  to  us  to  describe  straight 
lines,  but  at  other  times  the  apparent  paths  of  the  spots  are  some* 
what  elliptical,  and  they  present  the  greatest  curvature  about  the 
first  weeks  of  March  and  September. 

173.  Periodicity  in  the  number  of  the  solar  spots. — The  number 
of  the  solar  spots  varies  greatly  in  different  years.     Some  years 
the  sun's  disc  is  never  seen  entirely  free  from  spots,  while  in  other 
years,  for  weeks  and  even  months  together,  no  spots  of  any  kind 
can  be  perceived.     From  a  continued  series  of  observations,  em- 
bracing a  period  of  38  years,  it  appears  that  the  spots  are  subject 
to  a  certain  periodicity.     The  number  of  the  spots  increases  dur- 
ing 5  or  6  years,  and  then  diminishes  during  about  an  equal  pe- 
riod of  time,  the  interval  between  two  consecutive  maxima  being 
from  10  to  12  years. 

As  this  period  corresponds  to  the  time  of  one  revolution  of 
Jupiter,  it  suggests  the  idea  that  possibly  Jupiter  may  have  the 
power  of  sensibly  disturbing  the  sun's  surface. 

174.  The  sun  not  a  solid  body. — A  comparison  of  the  dark  lines 
in  the  solar  spectrum  has  led  to  the  conclusion  that  the  elements 
of  which  the  sun  is  composed  are  to  a  great  extent  the  same  as 
those  found  upon  the  earth.     The  existence  of  iron,  nickel,  and 
several  other  well-known  metals  in  the  sun's  atmosphere  is  con- 
sidered as  proved ;  and  since  the  density  of  the  sun  is  only  one 
fourth  that  of  the  earth,  while  the  force  of  gravity  is  28  times  its 
force  upon  the  earth,  we  can  not  suppose  that  any  large  part  of 
the  sun's  mass  is  in  the  condition  of  a  solid  or  even  a  liquid  body. 
The  most  refractory  substances,  iron  and  nickel,  exist  upon  the 
sun^in  the  state  of  elastic  vapor.     Hence  the  temperature  of  the 
sun's  surface  is  extremely  elevated,  far  beyond  the  heat  of  terres- 
trial volcanoes.     It  is  possible  that  the  centre  of  the  sun  consists 


PHYSICAL   CONSTITUTION  OF  THE  SUN. 


97 


of  matter  in  the  liquid  or  even  the  solid  state ;  but  it  is  probable 
that  the  principal  part  of  the  sun's  volume  consists  of  matter  in 
the  gaseous  condition. 

175.  Nature  of  the  sun's  photosphere. — The  bright  envelope  of 
the  sun,  which  we  call  its  photosphere,  consists  of  matter  in  a  state 
analogous  to  that  of  aqueous  vapor  in  terrestrial  clouds ;  that  is, 
in  the  condition  of  a  precipitate  suspended  in  a  transparent  at- 
mosphere. This  photosphere  is  not  only  intensely  luminous,  but 
intensely  hot,  and  the  thermoscope  indicates  that  it  radiates  more 
heat  than  the  solar  spots ;  but  this  does  not  prove  that  the  photo- 
sphere is  really  hotter  than  the  nucleus  of  a  solar  spot,  for  gases 
radiate  heat  more  feebly  than  solids  of  the  same  temperature. 
The  matter  of  the  photosphere  probably  consists  of  particles  pre- 
cipitated in  consequence  of  their  being  cooled  by  radiation. 

The  sun's  gaseous  envelope  extends  far  beyond  the  photo- 
sphere. During  total  eclipses  we  observe  protuberances  rising 
to  a  height  of  80,000  miles  above  the  surface  of  the  sun,  which 
requires  us  to  admit  the  existence  of  bodies  analogous  to  clouds 
floating  at  great  elevations  in  an  atmosphere ;  and  if  the  extent 
of  the  solar  atmosphere  compared  with  the  height  of  the  visi- 
Fig.so.  ble  clouds  corresponds  with 

what  exists  upon  the  earth, 
we  must  conclude  that  the 
solar  atmosphere  extends  to 
at  least  a  million  of  miles 
beyond  his  surface. 

176.  Nature  of  the  penum- 
bra. —The  penumbra  of  a 
solar  spot  appears  to  be 
formed  of  filaments  of  pho- 
tospheric  light  converging 
toward  the  centre  of  the 
nucleus,  each  of  the  fila- 
ments having  the  same  light 
as  the  photosphere,  and  the 
sombre  tint  results  from  the 
dark  interstices  between  the 
luminous  streaks,  as  in  a  steel  engraving  shades  are  produced  by 

G 


98  ASTRONOMY. 

dark  lines  separated  by  white  interstices.  The  convergence  of 
the  luminous  streaks  of  the  penumbra  toward  the  centre  of  the 
spot  indicates  the  existence  of  currents  flowing  toward  the  cen- 
tre. These  converging  currents  probably  meet  an  ascending  cur- 
rent of  the  heated  atmosphere,  by  contact  with  which  the  matter 
of  the  photosphere  is  dissolved,  and  becomes  non-luminous. 

The  faculse  are  ascribed  to  commotions  in  the  photosphere,  by 
which  the  thickness  of  the  phosphorescent  stratum  is  rendered 
greater  in  some  places  than  in  others,  and  the  surface  appears 
brightest  at  those  points  where  the  luminous  envelope  is  thickest. 

177.  Motion  of  the  solar  spots. — The  spots  are  not  stationary  on 
the  sun's  disc,  for  the  apparent  time  of  revolution  of  some  of  the 
spots  is  much  greater  than  that  of  others.     In  one  instance,  the 
time  of  the  sun's  rotation,  as  deduced  from  observations  of  a  solar 
spot,  was  only  24d.  7h.,  while  in  another  case  it  amounted  to  26d. 
6h.     This  difference  can  only  be  explained  by  admitting  that  the 
spots  have  a  motion  of  their  own  relative  to  the  sun's  surface, 
just  as  our  clouds  have  a  motion  relative  to  the  earth's  surface. 

The  motion  of  the  solar  spots  in  latitude  is  very  small,  and 
this  motion  is  sometimes  directed  toward  the  equator,  but  gener- 
ally from  the  equator.  The  motion  of  the  spots  in  longitude  is 
more  decided.  Spots  near  the  equator  have  an  apparent  move- 
ment of  rotation  more  rapid  than  those  at  a  distance  from  the 
equator.  While  at  the  equator  the  daily  angular  velocity  of  ro- 
tation is  865',  in  lat.  20°  it  is  only  840',  and  in  lat.  30°  it  is  816'. 
Hence  a  point  on  the  sun's  equator  makes  a  complete  rotation  in 
25  days,  but  a  point  in  lat.  30°  makes  one  rotation  in  26^-  days. 

178.  Cause  of  the  movements  of  the  solar  spots. — The  heat  of  the 
sun  must  be  continually  dissipated  by  radiation.    If  this  radiation 
is  more  obstructed  in  some  regions  than  in  others,  heat  must  ac- 
cumulate in  such  places.     Now  the  phenomena  observed  during 
total  eclipses  indicate  in  the  sun's  atmosphere  the  existence  of 
large  masses  analogous  to  terrestrial  clouds.     Wherever  these 
clouds  prevail,  the  free  radiation  of  heat  from  the  sun  must  be 
obstructed,  and  heat  must  rapidly  accumulate.     The  solar  atmos- 
phere tends  to  move  toward  these  heated  centres,  and  this  must 
be  accompanied  by  an  upward  motion  at  the  centre.    The  heated 
air  thus  ascending  partly  dissolves  and  partly  divides  the  matter 


PHYSICAL   CONSTITUTION   OF   THE   SUN.  99 

of  the  photosphere,  causing  it  to  heap  up  in  a  ring  around  the 
opening,  producing  thus  around  the  margin  of  the  penumbra  the 
appearance  of  a  border  of  light  more  intense  than  the  general 
photosphere. 

A  general  movement  of  the  atmosphere  toward  one  point  must 
create  a  tendency  to  revolve  around  this  centre,  for  the  same  rea- 
son that  terrestrial  storms  sometimes  rotate  about  a  vertical  axis. 
Such  a  motion  of  the  solar  spots  has  been  repeatedly  indicated 
rig.  5t.  by  observation.     Moreover,  solar  spots  have 

sometimes  exhibited  a  spiral  structure  such 
as  might  be  supposed  to  result  from  rotation 
about  a  vertical  axis.  Fig.  51  represents  such 
a  spot  observed  by  M.  Secchi  at  Home  in  1857. 
The  nucleus  exhibited  two  centres  perfectly 
black,  while  the  penumbra  showed  numerous 
dark  lines  extending  spirally  from  these  cen- 
tres, and  a  large  spiral  filament,  in  the  form  of  an  eagle's  beak, 
extended  far  within  the  nucleus. 

179.  Zodiacal  light. — The  zodiacal  light  is  a  faint  light,  some- 
what resembling  that  of  the  Milky  Way,  or  more  nearly  that  of 
the  tail  of  a  comet,  and  is  seen  at  certain  seasons  of  the  year  in 
the  west  after  the  close  of  twilight  in  the  evening,  or  in  the  east 
before  its  commencement  in  the  morning.  Its  apparent  form  is 
nearly  that  of  a  cone  with  its  base  toward  the  sun,  and  its  axis 
F,g  53  is  situated  nearly  in  the  plane  of  the 

ecliptic.  The  season  most  favorable 
for  observing  this  phenomenon,  is 
when  its  direction,  or  the  direction 
of  the  ecliptic,  is  most  nearly  perpen- 
dicular to  the  horizon.  For  places 
near  the  latitude  of  New  York,  this 
occurs  about  the  1st  of  March  for 
the  evening,  and  about  the  10th  of 
October  for  the  morning. 

The  distance  to  which  the  zodi- 
acal light  extends  from  the  sun,  va- 
ries from  20°  or  30°  to  80°  or  90°, 
and  sometimes  even  more  than  90°. 
Its  breadth  at  its  base  perpendicu- 
larly to  its  length,  varies  from  8°  to 


100  ASTRONOMY. 

30°.  It  is  brightest  in  the  parts  nearest  the  sun,  and  in  its  up- 
per  part  its  light  fades  away  by  insensible  gradations,  so  that  dif- 
ferent observers  at  the  same  time  and  place  assign  to  it  different 
limits.  Under  favorable  circumstances,  it  has  been  seen  to  extend 
entirely  across  the  heavens. 

It  is  probable  that  the  zodiacal  light  is  an  envelope  of  very  rare 
matter  surrounding  the  sun,  and  extending  beyond  the  orbits  of 
Mercury  and  Venus,  and  at  times  even  beyond  the  orbit  of  the 
earth.  If  the  sun  could  be  viewed  from  one  of  the  other  stars,  it 
would  probably  appear  to  be  surrounded  by  a  nebulosity,  similar 
to  that  in  which  some  of  the  fixed  stars  appear  to  be  enveloped, 
as  seen  from  the  earth. 


CHAPTER  VII. 

PRECESSION  OF  THE  EQUINOXES. — NUTATION. — ABERRATION.— 
LINE   OF  THE  APSIDES. 

179.  Fixed  position  of  the  ecliptic. — By  comparing  catalogues  of 
stars  formed  in  different  centuries,  we  find  that  the  latitudes  of  the 
stars  continue  always  nearly  the  same.    Hence  the  position  of  the 
ecliptic  among  the  stars  must  be  well-nigh  invariable. 

180.  Precession  of  the  equinoxes. — It  is  found  that  the  longitudes 
of  the  stars  are  continually  increasing,  at  the  rate  of  about  50"  in 
a  year.     Since  this  increase  of  longitude  is  common  to  all  the 
stars,  and  is  nearly  the  same  for  each  star,  we  can  not  ascribe  it  to 
motions  in  the  stars  themselves.     We  hence  conclude  that  the 
vernal  equinox,  the  point  from  which  longitude  is  reckoned,  has 
a  backward  or  retrograde  motion  along  the  ecliptic,  amounting  to 
50"  in  a  year,  while  the  inclination  of  the  equator  to  the  ecliptic 
remains  nearly  the  same.     This  motion  is  called  the  precession  of 
the  equinoxes,  because  the  place  of  the  equinox  among  the  stars 
each  year  precedes  (with  reference  to  the  diurnal  motion)  that 
which  it  had  the  previous  year. 

The  amount  of  precession  is  50".2  annually.  In  order  to  de- 
termine how  many  years  will  be  required  for  a  complete  revolu- 
tion of  the  equinoctial  points,  we  divide  1,296,000,  the  number  of 
seconds  in  the  circumference  of  a  circle,  by  50".2,  and  obtain 
25,800  years. 


PRECESSION  OF  THE   EQUINOXES.  101 

181.  The  pole  of  the  equator  revolves  round  the  pole  of  the  ecliptic. — 
Since  the  position  of  the  ecliptic  is  fixed,  or  nearly  so,  it  is  evi- 
dent that  the  equator  must  change  its  position,  otherwise  there 
could  be  no  motion  in  the  equinoctial  points;  and  a  motion  of 
the  equator  implies  a  motion  of  the  poles  of  the  equator.     Since 
the  obliquity  of  the  ecliptic  remains  nearly  constant,  the  distance 
from  the  pole  of  the  equator  to  the  pole  of  the  ecliptic  must  re- 
main nearly  constant;  and  we  may  conceive  the  phenomena  of 
precession  to  arise  from  the  revolution  of  the  pole  of  the  celestial 
equator  around  the  pole  of  the  ecliptic,  in  the  period  of  25,800 
years,  at  a  constant  distance  of  about  23f  degrees. 

182.  The  signs  of  the  zodiac  and  the  constellations  of  the  zodiac. — 
At  the  time  of  the  formation  of  the  first  catalogue  of  stars,  140 
years  before  Christ,  the  signs  of  the  ecliptic  corresponded  very 
nearly  to  the  constellations  of  the  zodiac  bearing  the  same  names. 
But  in  the  interval  of  2000  years  since  that  period,  the  vernal 
equinox  has  retrograded  about  28° ;  so  that  the  sign  Taurus  now 
corresponds  nearly  with  the  constellation  Aries,  the  sign  Gemini 
with  the  constellation  Taurus,  and  so  for  the  others. 

183.  The  pole  star  varies  from  age  to  age. — The  pole  of  the  equa- 
tor in  its  revolution  about  the  pole  of  the  ecliptic,  must  pass  in 
succession  by  different  stars.     At  the  time  the  first  catalogue  of 
the  stars  was  formed,  the  north  pole  was  nearly  12°  distant  from 
the  present  pole  star,  while  its  distance  from  it  is  now  less  than 
If  degrees.     The  pole  will  continue  to  approach  this  star  till  the 
distance  between  them  is  about  half  a  degree,  and  will  then  recede 
from  it.     After  a  lapse  of  about  12,000  years,  the  pole  will  have 
arrived  within  about  5°  of  a  Lyra,  the  brightest  star  in  the  north- 
ern hemisphere. 

184.  Cause  of  the  precession  of  the  equinoxes. — The  earth  may  be 
considered  as  a  sphere  surrounded  by  a  spheroidal  shell,  thick- 
est at  the  equator,  Art.  45.     The  matter  of  this  shell  may  be  re- 
garded as  forming  a  ring  round  the  earth,  in  the  plane  of  the  equa- 
tor.   Now  the  tendency  of  the  sun's  action  on  this  ring,  except  at 
the  time  of  the  equinoxes,  is  always  to  make  it  turn  round  the  in- 
tersection of  the  equator  with  the  ecliptic,  toward  the  plane  of  this 
latter  circle. 


102 


ASTRONOMY. 


The  solar  force  exerted  on  the  part  of  this  ring  that  is  above 
the  ecliptic,  may  be  resolved  into  two  forces,  one  of  which  is  in 
the  plane  of  the  equator,  and  the  other  perpendicular  to  it.  The 
latter  force  tends  to  impress  on  the  ring  a  motion  round  its  inter- 
section with  the  ecliptic.  So,  also,  the  solar  force  exerted  on  the 
part  of  the  ring  that  is  below  the  ecliptic,  may  be  resolved  into 
two,  one  in  the  plane  of  the  equator,  and  the  other  perpendicu 
lar  to  it.  The  sun's  attraction  upon  the  nearest  half  of  the  ring, 
tends  to  bring  the  plane  of  the  ring  nearer  to  the  plane  of  the 
ecliptic ;  while  its  attraction  upon  the  remoter  half  of  the  ring  pro- 
duces an  opposite  effect.  But  on  account  of  the  greater  distance, 
the  latter  effect  is  less  than  the  former;  so  that  the  ring  would 
turn  slowly  around  its  intersection  with  the  ecliptic,  and  the  two 
planes  would  ultimately  coincide,  were  it  not  for  the  operation  of 
another  cause. 

185.  How  to  find  the  resultant  of  two  rotary  motions. — While  the 
equatorial  ring  has  this  tendency  to  turn  about  the  line  of  the 
equinoxes,  it  also  rotates  on  an  axis  perpendicular  to  its  plane  in 
twenty-four  hours ;  that  is,  it  has  a  tendency  to  rotate  simultane- 
ously about  two  different  axes.  The  result  is  a  tendency  to  ro- 
tate about  an  intermediate  axis,  whose  position  is  determined  by 
the  following  theorem : 

rig.  54.  If  a  body  is  revolving  freely  round  the  axis  AB, 

•with  the  angular  velocity  V,  and  if  a  force  be  im- 
pressed upon  it  which  would  make  it  revolve  about 
the  axis  AC  with  an  an- 
gular velocity  V,  then 
the  body  will  not  revolve 
about  either  of  the  axes  AB,  AC,  but 
about  a  third  axis  AD,  situated  in  the 
plane  BAG,  and  the  angle  BAG  will  be 
divided  so  that 

sin.  BAD  :  sin.  CAD  : :  V :  V. 
Let  PP'  represent  the  axis  of  diurnal 

rotation    of 

the    equato-  s 

rial  ring,  and 

AB  the  line  of  the  equinoxes,  about 
which  it  also  tends  slowly  to  revolve. 


Pig.  55. 


PRECESSION   OF  THE  EQUINOXES.  103 


The  new  axis  of  rotation,  Ep',  will  be  situated  in  the  plane 
and  the  sine  of  its  angular  distance  from  each  of  the  former  axes 
will  be  in  the  inverse  ratio  of  the  angular  velocity  round  that 
axis.  Kepeating  the  same  construction  for  the  following  instant, 
we  shall  find  the  new  position  of  the  axis  will  be  Ep",  and  so  on  ; 
that  is,  the  pointy  will  be  made  to  describe  a  curve  around  C,  the 
pole  of  the  ecliptic. 

186.  Illustration  from  the  Gyroscope,  —  This  motion  of  the  earth's 
equatorial  ring  may  be  very  closely  imitated  by  a  modified  form 

of  the  gyroscope.  Let  AB  represent  a  brass  ring, 
supported  by  wires  AD,  BD,  which  are  connected 
with  an  axis,  DC,  whose  extremity  is  a  little  above 
the  centre  of  gravity  of  the  ring  AB,  and  rests 
upon  a  support,  CE.  When  the  ring  AB  is  at 
rest,  its  axis  DC  will  have  a  vertical  position.  If, 
however,  the  axis  be  inclined  from  the  vertical, 
and  be  made  to  rotate  by  twirling  it  with  the  fin- 
gers, the  plane  of  the  ring  will  turn  slowly  round 
in  azimuth,  preserving,  however,  a  nearly  constant  inclination  to 
the  horizon  ;  that  is,  the  axis  of  the  ring  will  describe  the  surface 
of  a  cone,  or  the  point  F  will  describe  the  circumference  of  a  cir- 
cle about  the  point  Gr. 

187.  Why  the  precession  is  so  slow.  —  If  the  earth  were  a  perfect 
sphere,  the  solar  forces  acting  on  the  opposite  hemispheres  would 
exactly  balance  one  another,  and  could  produce  no  displacement 
of  the  earth's  axis.    If  now  we  conceive  the  equatorial  ring  already 
described,  to  be  attached  to  the  spherical  part  of  the  earth,  which 
is  far  heavier  than  the  ring,  it  is  evident  that  the  ring,  having  to 
drag  around  with  it  this  great  inert  mass,  will  have  its  velocity  of 
retrogradation  proportionally  diminished.     Thus,  then,  the  entire 
globe  must  have  a  motion  similar  to  that  ascribed  to  the  ring,  but 
the  motion  will  be  extremely  slow. 

The  moon  produces  a  similar  retrogradation  in  the  intersection 
of  the  equator  with  the  plane  of  the  lunar  orbit,  but,  on  account 
of  its  nearness  to  the  earth,  its  effect  is  more  than  double  that  of 
the  sun.  The  planets  also,  by  their  attraction,  exert  a  small  in- 
fluence upon  the  position  of  the  equatorial  ring,  but  the  result  is 
slightly  to  diminish  the  amount  of  precession.  The  whole  effect 


104:  ASTRONOMY. 

of  the  sun  and  moon  is  50".37,  and  that  of  the  planets  0".16,  leav- 
ing the  actual  amount  of  precession  50".21  annually. 

Natation. 

188.  The  effect  of  the  action  of  the  sun  and  moon  upon  the 
earth's  equatorial  ring,  depends  upon  their  position  with  regard 
to  the  equator.  When  either  body  is  in  the  plane  of  the  equator, 
its  action  can  have  no  tendency  to  change  the  position  of  this 
plane,  and  consequently  none  to  change  the  positions  of  the  equi- 
noctial points.  Its  effect  in  producing  these  changes,  increases 
with  the  distance  of  the  body  from  the  equator,  and  is  greatest 
when  that  distance  is  greatest.  Twice  a  year,  therefore,  viz.,  at 
the  equinoxes,  the  effect  of  the  sun  to  produce  precession  is  noth- 
ing, while  at  the  solstices  the  effect  of  the  sun  is  a  maximum. 
On  this  account,  the  precession  of  the  equinoxes,  as  well  as  the 
obliquity  of  the  ecliptic,  is  subject  to  a  semi-annual  variation, 
which  is  called  the  solar  nutation.  There  is  also  an  inequality 
depending  upon  the  position  of  the  moon  which  is  called  lunar 
nutation.  The  maximum  value  of  the  lunar  nutation  in  longi- 
tude is  17".2,  and  that  of  the  solar  nutation  1".2. 

In  consequence  of  this  oscillatory  motion  of  the  equator,  Its 
pole,  in  revolving  about  the  pole  of  the  ecliptic,  does  not  move 
Fig  57  strictly  in  a  circle,  but  in  a  waving  curve, 

which  passes  alternately  within  and  with- 
out the  circle,  somewhat  similar  to  that  in 
Fig.  67. 


189.  Tropical  and  sidereal  years.  —  The 
time  occupied  by  the  sun  in  moving  from 
the  vernal  equinox  to  the  vernal  equinox 
again,  is  called  a  tropical  year. 
The  time  occupied  by  the  sun  in  moving  from  one  fixed  star 
to  the  same  fixed  star  again,  is  called  a  sidereal  year. 

On  account  of  the  precession  of  the  equinoxes,  the  tropical  year 
is  less  than  the  sidereal  year,  the  vernal  equinox  having  gone  west- 
ward so  as  to  meet  the  sun.  The  tropical  year  is  less  than  the 
sidereal  year,  by  the  time  that  the  sun  takes  to  move  over  50".2 
of  his  orbit.  This  amounts  to  20m.  22s. 

The  mean  length  of  a  tropical  year  expressed  in  mean  solar 
time  is  365d.  5h.  48m.  48s.  The  length  of  the  sidereal  year  is 
therefore  365d.  6h.  9m.  10s. 


ABERRATION.  105 


Aberration. 

190.  The  annual  motion  of  the  earth,  combined  with  the  mo- 
tion of  light,  causes  the  stars  to  appear  in  a  direction  different 
from  their  true  direction.  This  displacement  is  called  aberration. 
The  nature  of  this  effect  may  be  understood  from  the  following 
illustration  : 

If  we  suppose  a  shower  of  rain  to  fall  during  a  dead  calm  in 
vertical  lines,  if  the  observer  be  at  rest  the  rain  will  appear  to 
fall  vertically  ;  and  if  the  observer  hold  in  his  hand  a  tube  in  a 
vertical  position,  a  drop  of  rain  may  descend  through  the  tube 
without  touching  the  sides  ;  but  if  the  observer  move  forward, 
the  rain  will  strike  against  his  face  ;  and,  in  order  that  a  drop  of 
FJo.  ,£  rain  may  descend  through  the  tube  without 

,g'  touching  the  sides,  the  tube  must  be  inclined 
forward.  Suppose,  while  a  rain-drop  is  fall- 
ing from  E  to  D  with  a  uniform  velocity,  the 
spectator  moves  from  C  to  D,  and  carries  the 
tube  inclined  in  the  direction  EC.  A  drop 
of  rain  entering  the  tube  at  E,  when  the  tube 
has  the  position  EC,  would  reach  the  ground 
at  D  when  the  tube  has  come  into  the  posi- 


^ 

A   c     D  B  tion  FD  ;  that  is,  the  drop  of  rain  will  appear 

to  follow  the  direction  EC. 

Now  CD=EDxtang.  CED; 

that  is,  the  velocity  of  the  observer  =  velocity  of  the  rain  x  tan- 
gent of  the  apparent  deflection  of  the  rain-drop. 

191.  To  determine  the  amount  of  aberration.  —  The  aberration  of 
light  is  explained  in  a  similar  manner.  Let  AB  be  a  small  por- 
tion of  the  earth's  orbit,  and  S  the  position  of  a  star.  Let  CD  be 
the  distance  through  which  the  observer  is  carried  in  Is.,  and  ED 
the  distance  through  which  light  moves  in  Is.  If  a  straight  tube 
be  conceived  to  be  directed  from  the  eye  at  C  to  the  light  at  E, 
so  that  the  light  shall  be  in  the  centre  of  its  opening,  and  if  the 
tube  moves  with  the  eye  from  C  to  D,  remaining  constantly  par- 
allel to  itself,  the  light,  in  moving  from  E  to  D,  will  pass  along  the 
axis  of  the  tube,  and  will  arrive  at  D  when  the  earth  reaches  the 
same  point.  It  is  evident  that  the  star  will  appear  in  the  direc- 
tion of  the  axis  of  the  tube  ;  that  is,  the  star  appears  in  the  direc- 


106 

tion  S'D  instead  of  SD. 


ASTRONOMY. 


„«.*  „  ~  ...- The  velocity  of  the  earth  in  its  orbit  is 

19  miles  per  second ;  the  velocity  of  light  is  192,000  miles  per 
second. 
In  the  triangle  ECD,  we  have 


Fig.  53. 


Hence  CED=20"; 

that  is,  the  aberration  of  a  star  which  is  90°  from  the  path  in 
which  the  earth  is  moving,  amounts  to  20". 

192.  Effect  of  aberration  upon  a  star  situated  at  the  pole  of  t/ie 
ecliptic. — It  is  obvious  that  the  aberration  is  always  in  the  direc- 
tion in  which  the  earth  is  moving.     Its  effect,  therefore,  upon  the 
apparent  position  of  a  star,  will  vary  with  the  season  of  the  year 

Let  ABCD  represent  the  annual 
path  of  the  earth  around  the  sun  ; 
let  S  be  the  place  of  the  sun,  and 
s  the  place  of  a  star  so  situated  that 
the  line  85  is  perpendicular  to  the 
plane  of  the  ecliptic. 

When  the  er.rth  is  at  the  point 

A,  moving  toward  B,  the  aberration 
O  will  be  in  the  direction  sa  ;  that  is, 

the  star  appears  at  the  point  a. 
When  the  earth  has  arrived  at 

B,  the  aberration  will  be  in  the  di- 
rection sb  ;  that  is,  the  star  appears 

at  the  point  b. 

When  the  earth  has  arrived  at  C,  the  star  appears  at  the  point 
c;  and  when  the  earth  has  arrived  at  D,  the  star  appears  at  the 
point  d.  But  sa,  sb,  sc,  sd  are  each  20",  and  therefore  the  star 
will  appear  annually  to  describe  a  small  circle  in  the  heavens, 
40"  in  diameter. 

193.  Effect  upon  a  star  situated  in  the  plane  of  the  ecliptic. — If  the 
star  were  situated  in  the  plane  of  the  ecliptic,  in  the  direction  of 
the  line  AC  produced,  then,  when  the  earth  is  at  C,  the  aberration 
will  be  20",  as  before ;  but  when  the  earth  is  at  D,  the  aberration 
will  be  nothing,  because  the  earth  and  the  light  of  the  star  are 
moving  in  the  same  direction.     When  the  earth  is  at  A,  the  ab- 


LINE   OF  THE  APSIDES.  107 

erration  will  again  be  20'',  but  in  a  direction  opposite  to  what  it 
was  at  C ;  and  when  the  earth  is  at  B,  the  aberration  will  again 
be  nothing.  Hence  we  see  that  if  a  star  be  in  the  plane  of  the 
ecliptic,  it  will  appear  to  oscillate  to  and  fro  along  a  straight  line, 
20"  on  each  side  of  the  true  position  of  the  star,  and  this  line  will 
be  situated  in  the  plane  of  the  ecliptic. 

A  star  situated  between  the  ecliptic  and  its  poles,  will  appear 
annually  to  describe  an  ellipse  whose  major  axis  is  40",  but  its 
minor  axis  will  increase  with  its  distance  from  the  plane  of  the 
ecliptic. 

194.  The  apsides  of  the  earttis  orbit. — The  points  of  perihelion 
and  aphelion  of  the  earth's  orbit,  are  called  by  the  common  name 
of  apsides.     The  major  axis  of  the  earth's  orbit  is  therefore  called 
the  line  of  the  apsides. 

By  comparing  very  distant  observations,  it  is  found  that  the 
line  of  the  apsides  has  a  progressive  motion,  or  a  motion  east- 
ward amounting  to  about  12"  annually.  Since  the  equinox  from 
which  longitude  is  reckoned  moves  in  the  opposite  direction  50" 
annually,  the  longitude  of  the  perihelion  increases  about  62"  an- 
nually. 

At  this  rate,  the  line  of  the  apsides  would  complete  a  sidereal 
revolution  in  108,000  years,  or  a  tropical  revolution  in  20,900 
years.  For  the  cause  of  this  motion,  see  Arts.  279  478. 

195.  Changes  in  the  position  of  the  line  of  the  apsides. — The  line 
of  the  apsides,  thus  continually  moving  round,  must  at  one  period 
have  coincided  with  the  line  of  the  equinoxes.     The  longitude  of 
the  perihelion  in  1864  was  100°  16',  which  point  the  earth  passed 
on  the  1st  of  January.     The  time  required  to  move  over  an  arc 
of  100^°  at  the  rate  of  62"  annually,  is  about  5818  years,  which 
extends  back  nearly  4000  years  before  the  Christian  era — a  peri- 
od remarkable  for  being  that  to  which  chronologists  refer  the 
creation  of  the  world.     At  this  time  the  winter  and  spring  were 
equal,  and  longer  than  the  summer  and  autumn,  which  were  also 
equal. 

196.  Mean  place  and  true  place  ;  mean  anomaly  and  true  anoma~ 
ly. — The  mean  place  of  a  body  revolving  in  an  orbit,  is  the  place 
where  the  body  would  have  been  if  its  angular  velocity  had  been 


108  ASTRONOMY. 

uniform ;  the  true  place  of  a  body  is  the  place  where  the  body 
actually  is  at  any  time.  Equations  are  corrections  which  are  ap- 
plied to  the  mean  place  of  a  body,  in  order  to  get  its  true  place. 

The  angular  distance  of  a  planet  from  its  perihelion,  as  seen 
from  the  sun,  is  called  its  anomaly. 

If  an  imaginary  planet  be  supposed  to  move  from  perihelion  to 
aphelion  with  a  uniform  angular  motion  round  the  sun,  in  the 
same  time  that  the  real  planet  moves  between  the  same  points 
with  a  variable  angular  motion,  the  angular  distance  of  this  im- 
aginary planet  from  perihelion  is  called  its  mean  anomaly,  while 
its  actual  distance  at  the  same  moment  in  its  orbit  is  called  its 
true  anomaly. 

197.  Equation  of  the  centre. — The  difference  between  the  mean 
and  the  true  anomaly  is  called  the  equation  of  the  centre. 

Let  ABCD  be  the  orbit  of 
a  planet  having  the  sun  in 
one  of  the  foci  at  S.  With 
the  centre  S,  and  a  radius 
equal  to  the  square  root  of 
•JE  the  product  of  the  semi-axes 
of  the  ellipse,  describe  the 
circle  EBFD;  the  area  of 
this  circle  will  be  equal  to 
that  of  the  ellipse.  At  the 
same  time  that  a  planet  de- 
parts from  A,  the  perihelion,  to  describe  the  orbit  ABCD,  let  an 
imaginary  planet  start  from  E,  and  describe  the  circle  EBFD 
with  a  uniform  motion,  and  perform  a  whole  revolution  in  the 
same  period  that  the  planet  describes  the  ellipse.  The  imaginary 
planet  will  describe  around  S,  sectors  of  circles  which-  are  pro- 
portional to  the  times,  and  equal  to  the  elliptic  areas  described  in 
the  same  time  by  the  planet.  Suppose  the  imaginary  planet  to 
be  at  G;  then  take  the  sector  ASH  =  ESG,  and  H  will  be  the 
place  of  the  planet  in  the  ellipse.  The  angle  ESG-  is  called  the 
mean  anomaly ;  ASH  is  the  true  anomaly ;  and  GSH  is  the  equa- 
tion of  the  centre. 

If  we  consider  the  mean  and  the  true  anomaly  as  agreeing  at 
A,  the  angles  ESG  and  ASH  must  increase  unequally,  and  the 
true  anomaly  must  exceed  the  mean.  The  equation  of  the  cen- 


ANOMALISTIC  YEAR. — THE  MOON.  109 

£re  increases  till  the  planet  reaches  the  point  B.  From  B  to  C 
the  mean  anomaly  gains  upon  the  true,  until  at  C  they  coincide 
— that  is,  the  equation  of  the  centre  is  nothing.  Proceeding  from 
C,  the  mean  anomaly  must  exceed  the  true,  and  the  equation  of 
the  centre  increases  until  the  planet  reaches  the  point  D.  From 
D  to  A  the  true  anomaly  gains  upon  the  mean,  until  at  A  they 
coincide  again.  At  the  points  B  and  D  the  equation  of  the  cen- 
tre is  the  greatest  possible. 

The  greatest  value  of  the  equation  of  the  centre  for  the  sun  is 
1°  55'  27". 

198.  The  anomalistic  year. — The  time  occupied  by  the  earth  in 
moving  from  the  perihelion  to  the  perihelion  again,  is  called  the 
anomalistic  year.  This  period  must  be  a  little  longer  than  the 
sidereal  year,  since  the  earth  must  describe  a  further  arc  of  11".8 
before  reaching  the  perihelion ;  and  the  difference  will  be  equal 
to  the  time  necessary  for  the  earth  to  dgsmbe  11".8  of  its  orbit, 
or  4m.  35s.,  -which  gives  365d.  6h.  13m.  45s.  for  the  length  of  the 
anomalistic  year.  This  period  is  occasionally  used  in  astronom- 
ical investigations,  but  mankind  are  generally  more  concerned  in 
the  tropical  year,  on  which  the  return  of  the  seasons  depends. 


CHAPTER  VIII.  " 

THE  MOON  —  ITS  MOTION  —  PHASES  —  TELESCOPIC  APPEARANCE. 

199.  Distance  of  the  moon. — The  distance  of  the  moon  can  be 
computed  when  we  know  its  horizontal  parallax.     This  parallax 
varies  considerably  during  a  revolution  of  the  moon  round  the 
earth.     The  equatorial  parallax,  when  least,  is  53'  48",  and  when 
greatest,  61'  32".     The  mean  horizontal  parallax  of  the  moon  at 
the  equator  is  57'  2".3.     Hence  the  mean  distance  will  be  found 
by  the  proportion 

sin.  57'  2". 3  :  3963.35  : :  1 :  the  moon's  distance, 
which  is  found  to  be  238,885  miles. 

In  the  same  manner,  the  moon's  greatest  distance  is  found  to  be 
253,263  miles,  and  its  least  distance  221,436  miles. 

200.  Diameter  of  the  moon. — The  absolute  diameter  of  the  moon 
can  be  computed  when  we  know  its  apparent  diameter,  and  its 


110  ASTRONOMY. 

distance  from  the  earth.  The  apparent  diameter  varies  according 
to  its  distance  from  the  earth.  When  nearest  to  us,  it  is  33'  31  ".1 ; 
but  at  its  greatest  distance  it  is  only  29'  21'/.9.  At  its  mean  dis- 
tance the  apparent  diameter  is  31'  7".0.  Hence  the  absolute  di- 
ameter will  be  found  by  the  proportion 

1 :  238,885  ::  sin.  15'  33".5  :  the  moon's  semi-diameter, 
which  is  found  to  be  1081.1  miles.     Hence  the  moon's  diameter 
is  2162  miles. 

Since  spheres  are  as  the  cubes  of  their  diameters,  the  volume  of 
the  moon  is  Ty<h  that  of  the  earth.  Its  density  is  about  fths  (.615) 
the  density  of  the  earth,  and  its  mass  (=^x.615)  is  about  ^th 

of  the  mass  of  the  earth. 

. -frd-'*/-  6*  • 

201.  Definitions. — A  body  is  said  to  be  in  conjunction  with  the 
sun  when  its  longitude  is  the  same  as  that  of  the  sun ;  it  is  said 
to  be  in  opposition  to  the  sun  when  their  longitudes  differ  180° ; 
and  to  be  in  quadrature  when  their  longitudes  differ  90°  or  270°. 
The  term  syzygy  is  used  to  denote  either  conjunction  or  opposi- 
tion. 

The  octants  are  the  four  points  midway  between  the  syzygies 
and  quadratures. 

The  two  points  in  which  the  orbit  of  the  moon  or  a  planet  is 
cut  by  the  plane  of  the  ecliptic  are  called  nodes.  That  node  at 
which  the  body  passes  from  the  south  to  the  north  side  of  the 
ecliptic  is  called  the  ascending  nodc}  and  the  other  the  descending 
node. 

202.  Revolution  of  the  moon. — If  the  situations  of  the  moon  be 
observed  on  successive  nights,  it  will  be  found  that  it  changes  its 
position  among  the  stars,  moving  among  them  from  west  to  east ; 
that  is,  in  a  direction  contrary  to  that  of  the  diurnal  motion.     By 
this  motion  it  makes  a  complete  circuit  of  the  heavens  in  about 
27  days.    Hence  either  the  moon  revolves  round  the  earth,  or  the 
earth  round  the  moon.     Strictly  speaking,  the  earth  and  moon 
both  revolve  about  their  common  centre  of  gravity.     This  is  a 
point  in  the  line  joining  their  centres,  situated  at  an  average  dis- 
tance of  2690  miles  from  the  centre  of  the  earth,  or  about  1270 
miles  berieafif  the  surface  of  the  earth. 

203.  Sidereal  and  synodic  revolutions. — The  interval  of  time  oc- 


THE    MOON  —  ITS   MOTION.  Ill 

cupied  by  the  moon  in  performing  one  sidereal  revolution  round 
the  earth,  or  the  time  which  elapses  between  her  leaving  a  fixed 
star  until  she  again  returns  to  it,  is  27d.  7h.  43m.  11s. 

The  moon's  mean  daily  motion  is  found  by  dividing  360°  by 
the  number  of  days  in  one  revolution.  The  mean  daily  motion 
is  thus  found  to  be  13°.1764,or  about  13£  degrees. 

The  synodical  revolution  of  the  moon  is  the  interval  between 
two  consecutive  conjunctions  or  oppositions. 

The  synodical  revolution  of  the  moon  is  longer  than  the  side- 
real by  2d.  oh.  Om.  51s.?  which  is  the  time  required  by  that  body 
to  describe  with  its  mean  angular  velocity  of  13^  degrees  per  day 
the  arc  traversed  by  the  sun  since  the  previous  conjunction. 
Hence  we  find  the  duration  of  the  synodical  period  to  be  29d. 
12h.  44m.  2s. 

204.  How  the  si/nodical  period  is  determined.  —  The  mean  synod- 
ical period  may  be  determined  with  great  accuracy  by  observa- 
tions of  eclipses  of  the  moon.     The  middle  of  an  eclipse  is  very 
near  the  instant  of  opposition,  and  from  the  observations  of  the 
eclipse  the  exact  time  of  opposition  may  be  easily  computed. 
Now  eclipses  have  been  very  long  observed,  and  the  time  of  the 
occurrence  of  some  has  been  recorded  even  before  the  Christian 
era.     By  comparing  an  eclipse  observed  by  the  Chaldeans,  720 
B.C.,  with  recent  observations,  the  duration  of  the  mean  synodic 
period  has  been  ascertained  with  great  accuracy. 

205.  How  the  sidereal  period  is  derived  from  the  synodical.  —  The 
sidereal  period  may  be  deduced  from  the  synodical  as  follows: 

Let  P  =  the  length  of  the  sidereal  year, 

p  =  the  sidereal  revolution  of  the  moon, 
T  =  the  synodical  period  of  the  moon. 

Then  the  arc  which  the  moon  describes  in  order  to  come  into  con- 
junction writh  the  sun,  exceeds  360°  by  the  space  which  the  sun 
has  passed  over  since  the  preceding  conjunction.  This  excess 
is  found  by  the  proportion 


Then,  as  the  whole  distance  the  moon  must  move  from  the  sun 
to  reach  it  again,  is  to  one  circumference,  so  is  the  time  of  describ- 
ing the  former,  to  the  time  of  describing  the  latter;  that  is, 


112  ASTRONOMY. 


or  l  +  p:l::T: 

PT      365.25x29.53 
Whence       l> 


and  this  is  the  most  accurate  mode  of  determining  the  sidereal  pe- 
riod of  the  moon. 

206.  Moons  path.  —  The  moon's  observed  right  ascension  and 
declination  enable  us  to  determine  her  latitude  and  longitude.    By 
observing  the  moon  from  day  to  day  when  she  passes  the  merid- 
ian, we  find  that  her  path  does  not  coincide  with  the  ecliptic,  but 
is  inclined  to  it  at  an  angle  of  5°  8'  48",  and  intersects  the  eclip- 
tic in  two  opposite  points,  which  are  called  the  moon's  nodes. 

207.  Form  of  the  moons  orbit.  —  It  can  be  proved  in  a  manner 
similar  to  that  given  for  the  sun,  Arts.  Ill  and  11-1,  that  the  moon 
in  her  orbit  round  the  earth  obeys  the  following  laws  : 

1st.  The  moon's  path  is  an  ellipse,  of  which  the  earth  occupies 
a  focus. 

2d.  The  radius  vector  of  the  moon  describes  equal  areas  in  equal 
times. 

The  point  in  the  moon's  orbit  nearest  the  earth  is  called  her 
perigee,  and  the  point  farthest  from  the  earth  her  apogee.  The 
line  joining  the  apogee  and  perigee  is  called  the  line  of  the  apsides. 

208.  Eccentricity  of  the  moon's  orbit.  —  The  eccentricity  of  the 
lunar  orbit  may  be  found  by  observing  the  greatest  and  least  ap- 
parent diameters  of  the  moon,  in  the  same  manner  as  was  done 
in  the  case  of  the  sun,  Art  113. 

Example.  In  the  month  of  October,  1862,  the  greatest  apparent 
diameter  of  the  moon  was  33'  0".6,  and  the  least  was  29'  3-±".0. 
From  these  data  determine  the  eccentricity  of  the  lunar  orbit  dur- 
ing that  month. 

The  ratio  of  A  to  P  is       0.89569. 

Hence,  by  the  formula         e=~\  —  p> 

we  find  e—  0.0550,  or  about  -jL. 


THE   MOON — ITS  PHASES.  113 

209.  Interval  of  moon's  transits. — The  moon's  mean  daily  mo- 
tion in  right  ascension  is  13°. 17,  or  12°.19  greater  than  that  of 
the  sun.     Hence,  if  on  any  given  day  we  suppose  the  moon  to  be 
on  the  meridian  at  the  same  instant  with  the  sun,  on  the  next  day 
she  will  not  arrive  at  the  meridian  till  51m.  after  the  sun ;  that 
is,  the  interval  between  two  successive  meridian  passages  of  the 
moon  is,  on  the  average,  2-lh.  51m. 

In  consequence  of  the  inequalities  in  the  moon's  motion  in  right 
ascension,  this  interval  varies  from  2-lh.  38m.  to  25h.  6m. 

210.  Moon's  meridian  altitude. — The  moon's  altitude  when  it 
crosses  the  meridian  is  very  variable.     The  meridian  altitude  of 
the  sun  at  the  summer  solstice  is  46°  51'  (twice  the  obliquity  of 
the  ecliptic)  greater  than  it  is  at  the  winter  solstice.     Now,  since 
the  moon's  orbit  is  inclined  5°  9'  to  the  plane  of  the  ecliptic,  the 
moon  will  sometimes  be  distant  from  the  ecliptic  by  this  quanti- 
ty on  the  north  side,  and  at  other  times  by  the  same  quantity  on 
the  south  side ;  hence  the  greatest  meridian  altitude  of  the  moon 
will  exceed  its  least  by  46°  54' +10°  18',  or  57°  12'.     In  latitude 
41°  18',  the  greatest  meridian  altitude  of  the  moon  is  77°  18',  and 
its  least  20°  6'. 

211.  The  moon's  phases. — The  different  forms  which  the  moon's 
visible  disc  presents  during  a  synodic  revolution  are  called  phases. 

The  moon's  phases  are  completely  accounted  for  by  assuming 
her  to  be  an  opaque  globular  body,  rendered  visible  by  reflecting 
light  received  from  the  sun. 

Let  E  be  the  earth,  and  ABODE  the  orbit  of  the  moon,  the 
sun  being  supposed  to  be  at  a  great  distance  in  the  direction  AS. 
When  the  moon  is  in  conjunction  at  A,  the  enlightened  half  is 
turned  directly  from  the  earth,  and  she  must  then  be  invisible. 
It  is  then  said  to  be  new  moon. 

About  7£  days  after  new  moon,  when  she  is  in  quadrature  at 
C,  one  half  of  her  illumined  surface  is  turned  toward  the  earth, 
and  her  enlightened  disc  appears  as  a  semicircle.  She  is  then 
said  to  be  in  her^ir,^  quarter. 

About  15  days  after  new  moon,  when  she  is  in  opposition  at 
F,  the  whole  of  her  illumined  surface  is  turned  toward  the  earth, 
and  she  appears  as  a  full  circle  of  light.  It  is  then  said  to  be 
full  moon. 

H 


H4.  ASTRONOMY. 


C* 


About  7£  days  after  full  moon,  when  she  is  again  in  quadra- 
ture at  H,  one  half  of  her  UJumined  surface  being  turned  toward 
the  earth,  she  again  appears  as  a  semicircle.  She  is  then  said  to 
be  at  her  last  quarter. 

From  new  moon  to  first  quarter,  and  from  last  quarter  to  new 
moon,  her  enlightened  disc  is  called  a  crescent.  This  phase  is  rep- 
resented at  B  and  I.  The  two  extremities  of  the  crescent  are 
called  cusps,  or  horns.  From  first  quarter  to  full  moon,  and  from 
full  moon  to  last  quarter,  the  form  of  her  enlightened  disc  is  said 
to  be  gibbous.  This  phase  is  represented  at  D  and  GK  These 
phases  prove  conclusively  that  the  moon  shines  by  light  borrowed 
from  the  sun. 

The  interval  from  one  new  moon  to  the  next  new  moon  is 
called  a  lunation,  or  lunar  month.  It  is  evidently  the  same  as  a 
synodical  revolution  of  the  moon. 

212.  Obscure  part  of  the  moon's  disc. — When  the  moon  is  just  vis- 
ible after  new  moon,  the  whole  of  her  disc  is  quite  perceptible,  the 
part  not  fully  illumined  appearing  with  a  faint  light.  As  the 
moon  advances,  the  obscure  part  becomes  more  and  more  faint, 
and  it  entirely  disappears  before  full  moon.  This  phenomenon 
depends  on  light  reflected  from  the  earth  to  the  moon,  and  from 
the  moon  back  to  the  earth. 

When  the  moon  is  near  to  A,  she  receives  light  from  nearly 


THE  HARVEST  MOON.  115 

the  whole  of  the  earth's  illumined  surface,  and  this  light,  being  in 
part  reflected  back,  renders  visible  that  portion  of  the  disc  that  is 
not  directly  illumined  by  the  sun.  As  the  moon  advances  to* 
ward  opposition  at  F,  the  quantity  of  light  she  receives  from  the 
illumined  surface  of  the  earth  decreases ;  and  its  effect  in  render- 
ing the  obscure  part  visible,  is  farther  diminished  by  the  in- 
creased light  of  the  part  which  is  directly  illuminated  by  the 
sun's  rays. 

It  is  obvious  that,  to  an  observer  at  the  moon,  the  earth  must 
appear  as  a  splendid  moon,  presenting  all  the  phases  of  the  moon 
as  seen  from  the  earth,  and  having  more  than  three  times  its  ap- 
parent diameter. 

213.  Daily  retardation  of  the  moon's  rising  or  setting. — The  aver- 
age daily  retardation  of  the  moon's  rising  or  setting  is  the  same 
as  that  of  her  passage  over  the  meridian ;  but  the  actual  retarda- 
tion, being  affected  by  the  moon's  changes  in  declination,  as  well 
as  by  the  inequalities  of  her  motion  in  right  ascension,  is  subject 
to  greater  variation.     In  the  latitude  of  New  York,  the  least  daily 
retardation  is  23  minutes,  and  the  greatest  is  Ih.  17m. 

214.  Harvest  Moon. —  The  less  or  greater  retardation  of  the 
moon's  rising  attracts  most  attention  when  it  occurs  at  the  time 
of  full  moon.     When  the  retardation  has  its  least  value  near  the 
time  of  full  moon,  the  moon  rises  soon  after  sunset  on  several  suc- 
cessive evenings;  whereas,  when  the  retardation  is  greatest,  the 
moon  ceases  in  two  or  three  days  to  be  seen  in  the  early  part  of 
the  evening. 

When  the  moon  is  in  that  part  of  her  orbit  which  makes  the 
least  angle  with  the  horizon,  13  degrees  of  her  orbit  (which  is  her 
average  progress  in  a  day)  rises  above  the  horizon  at  New  York 
in  less  than  30  minutes.  This  happens  for  the  full  moon  near 
the  time  of  the  autumnal  equinox.  As  this  is  about  the  period 
of  the  English  harvest,  this  moon,  is  hence  called  the  Harvest 
Moon. 

215.  Effect  of  altitude  on  the  moon's  apparent  diameter. — The  ap- 
parent diameter  of  the  moon  is  not  the  same  at  the  same  instant 
for  all  points  of  the  earth,  on  account  of  their  different  distances 
from  the  moon.     As  the  moon  rises  above  the  horizon  (if  we  sup- 


116  ASTRONOMY. 

pose  its  distance  from  the  centre  of  the  earth  to  remain  constant), 
its  distance  from  the  place  of  observation  must  diminish,  while  its 
altitude  increases,  and,  consequently,  its  apparent  diameter  must 
increase.  This  effect  attains  its  maximum  when  the  moon  is  in 
the  zenith  of  the  spectator. 

The  distance  AB  is  about  equal  to  CB  or  CO,  and  exceeds  AG 

by  AC,  the  radius  of  the 

Fig.  61       TT/^S  J  ,  •    i     •         -L 

earth,  which  is  about  one 
sixtieth  of  the  moon's  dis- 
tance. Hence  the  angle 
GAH,  which  the  moou's 
radius  subtends  when  in 
the  zenith,  exceeds  the 
angle  BAD,  which  the 
moon's  radius  subtends 

when  in  the  horizon,  by  about  one  sixtieth  of  the  whole  quantity; 

that  is,  the  augmentation  of  the  moon's  diameter  on  account  of 

her  apparent  altitude  may  amount  to  more  than  half  a  minute. 
The  apparent  enlargement  of  the  moon  near  the  horizon  is  an 

optical  illusion,  as  explained  Art.  92. 

216.  Has  Hie  moon  an  atmosphere? — There  is  no  considerable  at- 
mosphere surrounding  the  moon.     This  is  proved  by  the  absence 
of  twilight.     Upon  the  earth,  twilight  continues  until  the  sun  is 
18°  below  the  horizon ;  that  is,  day  and  night  are  separated  by  a 
belt  1200  miles  in  breadth,  in  which  the  transition  from  light  to 
darkness  is  not  sudden,  but  gradual — the  light  fading  away  into 
the  darkness  by  imperceptible  gradations.     This  twilight  results 
from  the  refraction  and  reflection  of  light  by  our  atmosphere; 
and  if  the  moon  had  an  atmosphere,  we  should  notice,  in  like  man- 
ner, a  gradual  transition  from  the  bright  to  the  dark  portions  of 
the  moon's  surface.     Such,  however,  is  not  the  case.    The  bound- 
ary between  the  light  and  darkness,  though  irregular,  is  perfectly 
well  defined  and  sudden.    Close  to  this  boundary,  the  unillumined 
portion  of  the  moon  appears  just  as  dark  as  any  portion  of  the 
moon's  unillumined  surface. 

217.  Argument  from  the  absence  of  refraction. — The  absence  of  an 
atmosphere  is  also  proved  by  the  absence  of  refraction  when  the 
moon  passes  between  us  and  the  distant  stars.     Let  AB  represent 


LIGHT  OF  THE   FULL   MOON.  117 

Fig.  63.  the    disc    of   the 

moon,  and  CD  an 
^'  atmosphere  sup- 
posed to  surround 
it.  Let  SAE  rep- 
resent a  straight 

line  touching  the  moon  at  A,  and  proceeding  toward  the  earth, 
and  let  S  be  a  star  situated  in  the  direction  of  this  line.  If  the 
moon  had  no  atmosphere,  this  star  would  appear  to  touch  the  edge 
of  the  moon  at  A ;  but  if  the  moon  had  an  atmosphere,  this  at- 
mosphere would  refract  light ;  and  a  star  behind  the  edge  of  the 
moon  in  the  position  S'  would  be  visible  at  the  earth,  for  the  ray 
S'A  would  be  bent  by  the  atmosphere  into  the  direction  AE'. 
So,  also,  near  the  opposite  limb  of  the  moon,  a  star  might  be  seen 
at  the  earth,  although  really  behind  the  edge  of  the  moon.  Hence 
we  see  that  if  the  moon  had  an  atmosphere,  the  time  during  which 
a  star  would  be  concealed  by  the  moon  would  be  less  than  if  it 
had  no  atmosphere ;  and  the  amount  of  this  effect  must  be  pro- 
portional to  the  density  of  the  atmosphere. 

Many  thousand  occultations  of  stars  by  the  moon  have  been 
observed,  and  no  appreciable  effect  of  refraction  has  ever  been 
detected.  This  species  of  observation  is  susceptible  of  such  accu- 
racy, that  if  the  refraction  amounted  to  4"  of  arc,  it  is  believed  that 
it  could  not  fail  to  be  detected  in  the  mean  of  a  large  number  of 
observations.  Now  the  earth's  atmosphere  changes  the  direction 
of  a  ray  of  light  more  than  half  a  degree  when  it  enters  the  at- 
mosphere, and  the  same  when  it  leaves  it,  making  a  total  deflec- 
tion of  over  4000".  Hence  we  conclude  that  if  the  moon  have  an 
atmosphere,  its  density  can  not  exceed  one  thousandth  part  of  the 
density  of  our  own.  Such  an  atmosphere  is  more  rare  than  that 
which  remains  under  the  receiver  of  the  best  air-pump  when  it 
has  reached  its  limit  of  exhaustion. 

218.  Light  of  the  full  moon.— The  light  received  from  the  full 
moon  was  compared  by  Bouguer  with  the  light  received  from  the 
sun,  by  comparing  each  with  the  light  of  a  candle.  The  light  of 
the  sun  being  admitted  into  a  dark  room  through  a  small  aper- 
ture, he  placed  in  front  of  the  operator  a  concave  lens,  to  dimin- 
ish the  intensity  of  the  sun's  rays  by  causing  them  to  diverge. 
He  then  placed  a  candle  at  such  a  distance  that  its  light  received 


118  ASTRONOMY. 

upon  a  screen  was  exactly  equal  to  that  of  the  sun  received  upon 
the  same  screen. 

Repeating  this  experiment  at  night  with  the  full  moon,  he  com- 
pared the  light  of  the  moon  with  that  of  the  candle.  By  several 
experiments  of  this  kind,  he  arrived  at  the  conclusion  that  the  sun 
illumines  the  earth  300,000  times  more  than  the  full  moon. 

Professor  G.  P.  Bond  compared  the  light  of  the  moon  with  that 
of  the  sun  by  placing  in  the  sun's  light  a  glass  globe  with  a  sil- 
vered surface,  and  comparing  the  brightness  of  the  reflected  im- 
age of  the  sun  with  an  artificial  light,  and  afterward  comparing 
the  light  of  the  full  moon  with  the  same  standard.  He  hence  in- 
ferred that  the  light  of  the  sun  was  470,000  times  that  of  the  full 
moon. 

219.  Heat  of  the  moon. — Until  recently,  the  most  delicate  exper- 
iments had  failed  to  detect  any  heat  in  the  light  of  the  moon.    The 
light  of  the  full  moon  has  been  collected  into  the  focus  of  a  con- 
cave mirror  of  such  a  magnitude  as,  if  exposed  to  the  sun's  light, 
would  have  been  sufficient  to  evaporate  platinum ;  yet  no  sensi- 
ble effect  was  produced  upon  the  bulb  of  a  differential  thermom- 
eter so  delicate  as  to  show  a  change  of  temperature  amounting  to 
the  500th  part  of  a  degree.     This  experiment,  if  reliable,  would 
indicate  that  the  moon  reflects  a  less  proportion  of  the  heating 
rays  than  of  the  luminous  rays  of  the  sun. 

In  1846  Melloni  repeated  this  experiment  on  the  top  of  Mount 
Vesuvius  with  a  lens  of  three  feet  diameter,  and  found  feeble  in- 
dications of  heat  when  the  light  of  the  moon  was  concentrated 
upon  a  delicate  thermo-multiplier. 

In  the  summer  of  1856,  Professor  Smyth  repeated  this  experi- 
ment on  the  summit  of  Teneriffe,  over  10,000  feet  above  the  sea, 
and  found  that  the  heat  of  the  full  moon  was  equal  to  one  third 
that  of  an  ordinary  candle  placed  at  a  distance  of  15  feet. 

Even  this  small  amount  of  heat  appears  to  be  absorbed  by  the 
atmosphere  before  reaching  the  earth ;  and  near  the  earth's  sur- 
face, the  moon's  heat  is  inappreciable  by  the  most  delicate  means 
of  observation  hitherto  employed. 

220.  Telescopic  appearance  of  the  moon. — If  with  a  telescope  we 
examine  the  bounding  line  between  the  illumined  and  dark  por- 
tions of  the  moon's  surface,  especially  about  the  time  of  the  first 


THE   MOON— PARTICULAR  PHENOMENA  DESCRIBED.        119 

quarter,  we  shall  find  it  to  be  very  broken  and  irregular.  At 
some  distance  from  the  generally  illumined  surface  we  may  no- 
tice bright  spots,  often  entirely  surrounded  by  a  dark  ground ;  and 
we  also  find  dark  spots  entirely  surrounded  by  an  illumined  sur- 
face. These  appearances  change  sensibly  in  a  few  hours.  As  the 
light  of  the  sun  advances  upon  the  moon,  the  dark  spots  become 
bright ;  and  at  full  moon  they  all  disappear,  and  we  only  notice 
that  certain  regions  appear  more  dusky  than  others.  The  moon's 
surface  is  therefore  uneven ;  and,  by  observing  the  passage  of  the 
sun's  light  over  these  spots,  we  may  form  a  judgment  of  their  di- 
mensions and  figure. 

The  most  favorable  time  for  observing  these  inequalities  is  near- 
the  first  or  third  quarter,  because  then  the  shadows  of  the  mount- 
ains appear  of  their  greatest  length,  and  are  not  shortened  by  be- 
ing seen  obliquely.  See  Plate  II.,  Fig.  2,  which  gives  a  represent- 
ation of  a  small  portion  of  the  moon's  surface  as  seen  through  a 
powerful  telescope. 

221.  Particular  phenomena  described. — Near  the  bounding  line 
of  the  moon's  illumined  surface  we  frequently  observe  the  follow- 
ing phenomena :  A  bright  ring  nearly  circular ;  within  it,  on  the 
side  next  the  sun,  a  black  circular  segment ;  and  without  it,  on 
the  side  opposite  to  the  sun,  a  black  region  with  a  boundary  more 
or  less  jagged.    Near  the  centre  of  the  circle  we  sometimes  notice 
a  bright  spot,  and  a  black  stripe  extending  from  it  opposite  to  the 
sun.    After  a  few  hours,  the  black  portions  are  found  to  have  con- 
tracted in  extent,  and  in  a  day  or  two  entirely  disappear. 

After  about  two  weeks  these  dark  portions  reappear,  but  on 
the  side  opposite  to  that  on  which  they  were  before  seen ;  and 
they  increase  in  length  until  they  pass  entirely  within  the  dark 
portion  of  the  moon.  These  appearances  can  only  be  explained 
by  admitting  the  existence  of  a  circular  wall,  rising  above  the 
general  level  of  the  moon's  surface,  and  inclosing  a  large  basin, 
from  the  middle  of  which  rises  a  conical  peak. 

222.  Height  of  the  lunar  mountains. — If  the  distance  of  the  il- 
luminated summit  of  a  mountain  from  the  enlightened  part  of  the 
disc  be  measured  with  a  micrometer,  and  the  positions  of  the  sun 
and  moon  at  the  time  be  obtained  by  observation  or  computation, 
the  height  of  the  mountain  may  be  computed. 


120  ASTRONOMY. 

Let  AFE  be  the  illuminated  hemisphere 
^Fig.M.  of  the  moon,  SA  a  ray  of  the  sun  touch- 

mgl  ^\  &  ing  the  moon  at  A,  and  let  BD  be   a 

^^  \  mountain  so  elevated  that  its  summit  just 

t-      H£         \—          reaches  to  the  ray  SAB,  and  is  illumined 
J  while  the  intervening  space  AB  is  dark. 

/  Suppose  now  the  earth  to  be  in  the  direc- 

tion of  the  diameter  AE  produced.     Let 
the  angle  which  AB  subtends  at  the  earth 

be  measured  with  a  micrometer;  then,  since  the  distance  of  the 
moon  from  the  earth  is  known,  the  absolute  length  of  AB  can  be 
computed.  Then,  in  the  right-angled  triangle  ABC,  AC,  the  ra- 
dius of  the  moon,  is  known,  whence  BC  can  be  computed ;  and 
subtracting  AC  from  BC,  gives  BD,  the  height  of  the  mountain. 

If  the  earth  is  so  situated  that  the  line  AB  is  not  seen  perpen- 
dicularly, since  we  know  the  relative  positions  of  the  sun  and 
moon,  we  can  determine  the  inclination  at  which  AB  is  seen,  and 
hence  the  absolute  length  of  AB. 

The  height  of  a  mountain  may  also  be  computed  from  the 
measured  length  of  the  shadow  it  casts. 

The  greatest  elevation  of  any  lunar  mountain  which  has  been 
observed  is  23,800  feet.  The  altitudes  of  the  higher  mountains 
in  the  moon  are  probably  as  accurately  known  as  those  of  the 
highest  mountains  on  the  earth. 

223.  Circular  craters. — Mountain  ranges,  approaching  nearly  to 
the  form  of  circles,  are  very  common  on  the  moon's  surface.     They 
sometimes  have  a  diameter  of  over  50  miles,  and  a  height  of  2  or 
3  miles.     Tycho,  Kepler,  and  Copernicus  are  among  the  most  re- 
markable of  these  mountain  ranges.     See  Plate  II.,  Fig.  1.     Ty- 
cho, No.  1,  is  near  the  moon's  southern  limb ;  Kepler,  No.  2,  near 
the  eastern  limb ;  and  Copernicus,  No.  3,  a  little  west  of  Kepler. 
These  circular  mountains  bear  an  obvious  analogy  to  the  volcanic 
craters  upon  the  earth. 

224.  The  crater  of  Kilauea,  on  one  of  the  Sandwich  Islands,  is  a 
vast  basin,  more  than  three  miles  in  its  longer  diameter,  and  near 
ly  1000  feet  deep.     From  the  bottom  of  the  basin  rise  numerous 
little  cones,  from  which  smoke  is  almost  constantly  emitted,  and 
sometimes  melted  lava.     The  craters  of  most  volcanoes  exhibit 


THE   MOON— VOLCANOES.  121 

an  irregular  circular  wall  of  considerable  height,  sometimes  2  or 


Fig.  65. 


3  miles,  and  within  this  wall  rise  one  or  two  cones  formed  by  the 
occasional  overflowing  of  the  lava. 

225.  Lunar  volcanoes  compared  with  terrestrial. — The  lunar  vol- 
canoes differ  from  the  terrestrial  in  their  enormous  dimensions 
and  immense  number.     This  may  be  due,  in  some  degree,  to  the 
feeble  attraction  of  the  moon,  since  objects  on  the  moon's  surface 
weigh  only  one  sixth  what  they  would  on  the  earth. 

226.  It  is  certain  that  most  of  the  lunar  volcanoes  are  entirely 
extinct;  and  it  is  doubted  whether  any  signs  of  eruption  have  ever 
been  noticed.     The  spot  called  Aristarchus,  marked  4  on  Fig.  1, 
Plate  II.,  is  so  brilliant  that  some  have  concluded  it  to  be  an  active 
volcano.     Herschel  observed  on  the  dark  portion  of  the  moon 
three  bright  points,  which  he  ascribed  to  volcanic  fires ;  but  the 
same  lights  may  be  seen  every  month,  and  they  are  probably  to 
be  ascribed  to  mountain  peaks  which  have  an  unusual  power  of 
reflecting  the  feeble  light  which  is  emitted  by  the  earth.     It  is 
believed  that  all  the  inequalities  of  brightness  observed  on  the 
moon's  surface  (with  the  exception  of  the  shadows  described  in 
Arts.  220-1)  result  from  a  difference  in  the  nature  of  the  reflect- 
ing materials.     Two  distinguished  astronomers,  Beer  and  Madler, 
who  have  studied  the  moon's  surface  with  greater  care  than  any 
one  else,  assert  that  they  have  never  seen  any  thing  that  could  au- 


J22  ASTRONOMY. 

thorize  the  conclusion  that  there  are  in  the  moon  volcanoes  now 
in  a  state  of  ignition. 

227.  Streaks  of  light  from  Tycho. — Very  remarkable  streaks  of 
light  are  seen  diverging  from  several  of  the  lunar  craters.    These 
are  quite  conspicuous  about  Tycho,  Kepler,  Copernicus,  and  Aris- 
tarchus.    One  of  these  streaks  of  light  diverging  from  Tycho  can 
be  traced  1700  miles.     These  streaks  cross  ridges  and  valleys 
without  interruption ;  and  some  of  them  have  been  noticed  to 
cast  shadows.     They  are  thought  to  have  resulted  from  some  vi- 
olent volcanic  eruption,  by  which  enormous  crevices  were  opened 
in  the  moon's  surface.     These  crevices  are  supposed  to  have  been 
filled  with  melted  lava,  which  congealed  into  a  glassy  rock,  hav- 
ing a  more  brilliant  reflecting  surface  than  the  general  disc  of  the 
moon.     Similar  phenomena,  but  upon  a  far  less  extensive  scale. 
have  taken  place  on  the  earth's  surface. 

228.  There  is  no  water  on  the  moon's  surface.     The  dusky  re- 
gions, which,  were  once  supposed  to  be  seas,  are  regions  compara- 
tively level ;  but  upon  which,  with  a  good  telescope,  we  can  de- 
tect black  shadows,  indicating  the  existence  of  permanent  ine- 
qualities, which  could  not  exist  on  a  fluid  surface.     Moreover,  if 
there  were  any  water  on  the  moon's  surface,  a  portion  of  it  would 
rise  in  vapor,  and  form  an  atmosphere  which  would  refract  light 
to  an  extent  far  beyond  what  we  actually  observe. 

229.  Can  volcanoes  exist  without  air  or  water? — It  may  be  ob- 
jected that  volcanoes  could  not  exist  without  air  or  water.     It  is 
not  certain  that  the  presence  of  air  is  necessary  to  the  activity  of 
a  volcano.     Volcanoes  may  be  ascribed  to  the  primitive  heat  of 
the  globe,  or  to  galvanic  action  on  a  large  scale.     A  commotion 
of  the  melted  lava  would  be  instantly  produced  by  the  introduc- 
tion of  water,  which  would  suddenly  generate  large  quantities  of 
steam  ;  and  it  might  also  be  produced  by  the  presence  of  various 
other  bodies ;  as,  for  example,  sulphur,  which  almost  invariably 
accompanies  volcanic  eruptions.     Some  similar  substance  might 
cause  an  eruption  of  a  lunar  volcano  without  the  agency  of  water. 

230.  Can  animal  life  exist  upon  the  moon  ? — Air  and  water  are 
necessary  to  the  support  of  both  animal  and  vegetable  life.     It  is 


THE   MOON— DOES  IT  INFLUENCE  THE  WEATHER?        123 

doubtful,  therefore,  whether  even  the  humblest  form  of  life  with 
which  we  are  acquainted  could  exist  upon  the  moon.  Nothing 
has  ever  been  discovered  upon  the  moon's  surface  to  indicate  the 
agency  of  human  beings,  or  the  presence  of  any  form  of  animal 
or  vegetable  life.  The  extremes  of  temperature  upon  the  moon's 
surface  must  be  far  more  violent  than  they  are  upon  the  earth. 
For  14  successive  days  the  sun  shines  uninterruptedly  upon  the 
same  portion  of  the  moon,  and  for  the  next  14  days  his  light  is 
entirely  withdrawn.  During  the  first  period,  the  moon's  surface 
must  become  intensely  heated ;  and  during  the  next  fortnight  the 
cold  must  be  equally  severe,  since  there  is  no  atmosphere  or  clouds 
to  obstruct  the  radiation  of  heat. 

While,  then,  we  are  compelled  to  say  that  Infinite  wisdom  and 
power  can  create  beings  to  live  in  such  a  world,  we  can  safely 
assert  that  no  varieties  of  animal  or  vegetable  life  with  which  we 
are  acquainted  can  exist  in  the  moon. 

231.  Does  the  moon  influence  the  weather? — The  effect  of  the 
moon  upon  the  weather  is  very  slight,  and  can  only  be  detected  by 
taking  the  mean  of  a  long-continued  series  of  accurate  observa- 
tions.    From  a  comparison  of  observations  of  forty  years,  it  has 
been  concluded  that  there  is  a  monthly  fluctuation  of  temperature 
amounting  to  about  two  degrees  of  Fahrenheit,  which  is  due  to  the 
moon's  influence,  the  maximum  occurring  six  or  seven  days  aft- 
er new  moon,  and  the  minimum  about  four  days  after  full. 

The  moon  also  appears  to  exert  a  slight  influence  upon  the 
cloudiness  of  the  sky,  as  well  as  upon  the  number  of  rainy  days, 
the  greatest  cloudiness  occurring  near  the  third  quarter,  and  the 
least  cloudiness  about  the  time  of  new  moon,  the  fluctuation 
amounting  to  four  per  cent,  of  the  entire  sky. 

Although  the  preceding  results  seem  to  be  well  established,  it 
will  be  noticed  that  the  effect  ascribed  to  the  moon  is  quite  small, 
so  that  to  ordinary  daily  observation  this  influence  is  lost  in  the 
irregular  fluctuations  of  the  weather,  which  are  due  to  other  causes. 

232.  Does  the  moon  influence  the  pressure  of  the  air? — Many  have 
imagined  that  inasmuch  as  the  moon  elevates  the  water  of  the 
ocean,  its  disturbing  influence  ought  to  be  much  greater  upon  a 
fluid  of  such  mobility  as  our  atmosphere.    The  moon  does  indeed 
influence  the  pressure  of  the  air,  but  its  disturbing  force  is  ex- 
tremely small.     At  Singapore,  under  the  equator,  when  the  moon 


124  ASTRONOMY. 

is  on  the  meridian,  the  barometer  is  higher  by  -nnnrth  of  an  inch 
than  when  the  moon  is  six  hours  from  the  meridian ;  at  St.  He- 
lena, in  Lat.  15°  55',  this  difference  amounts  to  -nnnrth  of  an  inch ; 
and  in  our  latitude  the  difference  should  be  still  less.  This  effect 
is  so  minute  that  it  can  only  be  detected  by  the  most  accurate  ob- 
servations, continued  for  a  period  of  several  years.  Indeed,  it  has 
never  been  shown  that  the  moon  exerts  any  influence  upon  the 
weather,  except  that  which  is  of  the  feeblest  kind,  and  which  is 
only  appreciable  after  a  very  long  series  of  the  best  observations. 

233.  Moon's  rotation  upon  an  axis. — The  various  spots  on  the 
moon  always  occupy  nearly  the  same  positions  upon  the  disc,  from 
which  it  follows  that  nearly  the  same  surface  is  always  turned  to- 
ward the  earth.     Hence  we  conclude  that  the  moon  rotates  upon 
an  axis  in  the  same  time  that  she  makes  a  revolution  in  her  orbit. 
If  the  moon  had  no  motion  of  rotation,  then  in  opposite  parts  of 
her  orbit  she  would  present  opposite  sides  to  the  earth.    In  order 
that  a  globe  which  revolves  in  a  circle  around  a  centre  should 
turn  continually  the  same  hemisphere  toward  that  centre,  it  is 
necessary  that  it  should  make  one  rotation  upon  its  axis  in  the 
time  it  takes  to  revolve  about  the  centre. 

234.  Libralions  of  the  moon. — Although  it  is  true  that  nearly 
the  same  hemisphere  of  the  moon  is  always  turned  toward  the 
earth,  yet  the  moon  has  apparently  a  slight  oscillatory  motion, 
which  allows  us  to  see  a  portion  of  the  opposite  hemisphere. 
This  oscillatory  motion  is  called  libration. 

Libration  in  longitude. — While  the  moon's  angular  velocity  on 
its  axis  is  rigorously  uniform  throughout  the  month,  its  angular 
velocity  in  its  orbit  is  not  uniform,  being  most  rapid  when  nearest 
the  earth.  Hence  we  see  at  one  time  a  little  more  of  the  eastern 
or  western  edge  of  the  moon  than  we  do  at  another  time.  This 
is  called  the  libration  in  longitude. 

Libration  in  latitude. — The  axis  of  the  moon  is  not  quite  per- 
pendicular to  the  plane  of  her  orbit,  but  makes  an  angle  with  it 
of  83£  degrees.  On  account  of  this  inclination,  the  northern  and 
southern  poles  of  the  moon  incline  alternately  6£°  to  and  from 
the  earth.  "When  the  north  pole  leans  toward  the  earth,  we  see  a 
little  more  of  that  region,  and  a  little  less  when  it  leans  the  con- 
trary way.  This  variation  is  called  the  libration  in  latitude. 


THE   MOON — LUXAB   DAY.  125 

Diurnal  libration. — By  the  diurnal  motion  of  the  earth,  we  are 
carried  with  it  round  its  axis;  and  if  the  moon  presented  exactly 
the  same  hemisphere  toward  the  earth's  centre,  the  hemisphere 
visible  to  us  when  the  moon  rises,  would  be  different  from  that 
which  would  be  visible  to  us  when  the  moon  sets.  This  is  an- 
other cause  of  a  variation  in  the  edges  of  the  moon's  disc,  and  is 
called  the  diurnal  libration. 

In  consequence  of  all  these  librations,  we  can  see  somewhat 
more  than  half  of  the  surface  of  the  moon ;  yet  there  remains 
about  |ths  of  its  surface  which  is  always  hidden  from  our  view. 

235.  Lunar  day. — The  rotation  of  the  moon  upon  its  axis,  be- 
ing equal  to  that  of  its  revolution  in  its  orbit,  is  27^  days.     The 
intervals  of  light  and  darkness  to  the  inhabitants  of  the  moon,  if 
there  were  any,  would  be  altogether  different  from  those  upon  the 
earth.     There  would  be  about  35-i  hours  of  continued  light,  alter- 
nating with  354  hours  of  continued  darkness.    The  heavens  would 
be  perpetually  serene  and  cloudless.    The  stars  and  planets  would 
shine  with  extraordinary  splendor  as  well  in  the  day  as  in  the 
night.     The  inclination  of  her  axis  being  small,  there  would  be 
no  sensible  change  of  seasons.    The  inhabitants  of  one  hemisphere 
could  never  see  the  earth ;  while  the  inhabitants  of  the  other 
would  have  it  constantly  in  their  firmament  by  day  and  by  night, 
and  always  nearly  in  the  same  position.     To  those  who  inhabit 
the  central  part  of  the  hemisphere  presented  to  us,  the  earth  would 
appear  stationary  in  the  zenith,  with  the  exception  of  the  small 
effect  due  to  libration. 

The  earth  illumined  by  the  sun  would  appear  as  the  moon  does 
to  us,  but  with  a  superficial  magnitude  about  fourteen  times  as 
great.  Its  phases  would  also  be  similar  to  those  which  we  see  in 
the  moon. 

236.  Equality  of  the  periods  of  rotation  and  revolution.— That  the 
moon  should  rotate  on  an  axis  in  exactly  the  same  time  that  is 
required  for  a  revolution  around  the  earth,  can  not  be  supposed 
to  be  accidental. 

We  are  forced,  then,  to  seek  for  some  physical  cause  to  explain 
this  coincidence.  If  we  admit  that  originally  these  two  motions 
were  nearly  equal,  the  exact  equality  may  be  explained  as  follows : 
The  moon,  like  tho  earth,  was  probably  once  in  a  plastic  condi- 


126  ASTRONOMY. 

tion.  The  earth  would  then  act  upon  the  moon  as  the  moon  acts 
upon  the  earth  in  raising  the  tides,  only  with  much  greater  power; 
that  is,  if,  would  give  the  moon  an  elongated  figure,  its  major  axis 
pointing  toward  the  centre  of  the  earth.  If  the  moon  has  such 
an  elongated  figure,  the  earth  must  act  upon  it  as  upon  a  pendu- 
lum. When  a  pendulum  is  deflected  from  the  vertical  position, 
the  earth's  attraction  brings  it  back  again,  causing  it  to  oscillate 
to  and  fro.  So,  also,  if  the  longer  axis  of  the  moon  were  deflected 
from  pointing  toward  the  earth,  the  earth's  attraction  would  tend 
to  bring  it  back  to  this  position,  thus  tending  to  establish  a  rigor- 
ous equality  between  the  times  of  rotation  and  revolution  of  the 
moon. 

237.  Position  of  the  moon's  centre  of  gravity. — From  a  careful 
study  of  the  moon's  motions,  Hansen  concludes  that  the  centre 
of  gravity  of  the  moon  does  not  coincide  with  its  centre  of  figure, 
and  that  the  centre  of  figure  is  nearer  to  us  by  33  miles  than  the 
centre  of  gravity ;  in  other  words,  the  hemisphere  which  is  turned 
toward  the  earth  is  lighter  than  the  opposite  hemisphere,  and  may 
be  regarded  as  an  enormous  mountain,  rising  33  miles  above  the 
mean  level  of  the  moon.    This  lightness  may  be  the  result  of  vol- 
canic energy,  upheaving  the  crust,  and  leaving  large  cavities  be- 
neath ;  and  these  cavities  must  be  mainly  on  the  side  of  the  moon 
which  is  turned  toward  the  earth.     This  cause  may  have  contrib- 
uted to  produce  that  elongated  figure  of  the  moon  which  enables 
us  to  explain  the  exact  equality  between  the  time  of  rotation  upon 
its  axis  and  of  revolution  about  the  earth.     This  conclusion  of 
Hansen  is  not  accepted  by  all  astronomers. 

238.  Path  of  the  moon  in  its  motion  about  the  sun. — While  the 
moon  revolves  about  the  earth,  it  also  accompanies  the  earth  in 
its  motion  about  the  sun.     The  actual  path  described  by  the 
moon  will  then  be  an  undulating  line,  alternately  within  and  with- 
out the  orbit  of  the  earth.     The  undulations  are,  however,  so 
small,  in  comparison  with  the  dimensions  of  the  earth's  orbit,  that 
the  path  of  the  moon  is  always  concave  toward  the  sun.     The 
distance,  AB,  passed  over  by  the  earth  in  a  fortnight,  is  about  24 
millions  of  miles.     If  we  draw  a  chord  connecting  these  points, 
this  chord,  at  its  middle  point,  will  fall  about  700,000  miles  within 
the  orbit  of  the  earth,  while  the  greatest  distance  of  the  moon  from 


THE   MOON — CHANGES  OF  ITS  ORBIT.  127 

fig.  66. 


the  earth  is  only  253,000  miles.  The  moon's  path,  therefore,  ap- 
proaches so  near  to  that  of  the  earth  as  to  be  always  concave  to- 
ward the  sun. 

239.  Changes  of  the  moon's  orbit. — The  elliptic  path  described 
by  the  moon,  changes  gradually  from  month  to  month  both  in 
form  and  position.     Its  eccentricity  varies  within  certain  limits, 
being  sometimes  as  great  as  0.065,  and  sometimes  as  small  as 
0.049.     Its  mean  value  is  0.05484,  or  about  ^th. 

The  major  axis  of  the  moon's  orbit  is  not  fixed,  but  has  a  di- 
rect motion  on  the  ecliptic  at  the  rate  of  about  41°  in  a  year,  ac- 
complishing a  complete  revolution  in  a  little  less  than  nine  years ; 
so  that  in  4^-  years  the  perigee  arrives  where  the  apogee  was  be- 
fore. This  motion  of  the  line  of  the  apsides  is  not  equable 
throughout  the  whole  of  a  lunar  month  ;  for  when  the  moon  is  in 
syzygies,the  line  of  apsides  advances  in  the  order  of  the  signs,  but 
is  retrograde  in  quadratures.  The  direct  motion  is,  however, 
greater  than  the  retrograde. 

240.  Motion  of  the  line  of  the  nodes. — The  line  in  which  the  plane 
of  the  moon's  orbit  cuts  the  ecliptic,  is  called  the  line  of  the  nodes. 
The  position  of  the  nodes  is  found  by  observing  the  longitude  of 
the  moon  when  she  has  no  latitude ;  and  it  appears,  by  a  compari- 
son of  such  observations,  that  the  line  of  the  nodes  is  not  fixed, 
but  has  a  slow  retrograde  motion  at  the  rate  of  about  19°  in  a 
year.    By  this  motion  the  nodes  make  a  mean  tropical  revolution 
in  18  years  and  224  days,  nearly.     It  is  not,  however,  an  equable 
motion  throughout  the  whole  of  the  moon's  revolution.    The  node 
is  generally  stationary  when  the  moon  is  in  quadrature,  or  in  the 
ecliptic ;  in  all  other  parts  of  the  orbit  it  has  a  retrograde  motion, 
which  is  greater  the  nearer  the  moon  is  to  the  syzygies,  or  the 
greater  the  distance  from  the  ecliptic. 

Thus  we  see  that  the  path  of  the  moon  does  not  return  into  it- 


128 


ASTRONOMY. 


self,  but  is  a  curve  of  the  most  complicated  kind,  whose  form  and 
position  are  both  in  a  state  of  continual  change. 

241.  The  lunar  cycle. — The  lunar  cycle  consists  of  235  sy  nodical 
revolutions  of  the  moon,  which  differ  from  19  years  of  365^  days 
only  by  about  an  hour  and  a  half. 

For  29.5305887  x  235  =  6939.688  days. 

And  365^x19  =  6939.75    days. 

If,  then,  full  moon  should  happen  on  the  1st  of  January  in  the 
first  year  of  the  cycle,  it  will  happen  on  that  day  (or  within  a  very 
short  time  of  its  beginning  or  ending)  again  after  a  lapse  of  19 
years;  and  all  the  full  moons  in  the  interval  will  occur  on  the 
same  days  of  the  month  as  in  the  preceding  cycle.  This  period 
of  19  years  is  sometimes  called  the  Metonic  Cycle,  and  the  year  of 
the  Metonic  cycle  is  called  the  Golden  Number.  This  cycle  of  19 
years  is  used  for  finding  Easter.  Easter  day  is  the  first  Sunday 
after  the  full  moon  which  happens  upon  or  next  after  the  21st 
day  of  March.  The  present  lunar  cycle  began  in  1862,  when  full 
moon  occurred  April  14th.  Full  moon  also  occurred  on  the  same 
day  of  April  in  1843, 1824,  etc. 

The  following  are  the  dates  of  the  full  moons  next  following 
the  vernal  equinox  for  several  lunar  cycles: 


Year. 

Year.      I      Year. 

Yenr. 

Fnll  Moon. 

GoMen  No. 

1805 
1806 
1807 
etc. 

1824 
1825 
1826 
etc. 

1843 
1844 
1845 
etc. 

1862 
1863 
1864 
etc. 

April  14 
April  3 
March  22 

1 

2 
3 

etc. 

CHAPTER  IX. 

CENTRAL  FORCES.  — LAW   OF  GRAVITATION.  —  LUNAR  IRREGU- 
LARITIES. 

242.  Curvilinear  motion. — If  a  body  at  rest  receive  an  impulse 
in  any  direction,  it  will,  if  entirely  at  liberty  to  obey  that  impulse, 
move  in  that  direction,  and  with  a  uniform  rate  of  motion.  When 
a  body  moves  in  a  curve  line,  there  must  then  be  some  force  which 
at  every  instant  deflects  it  from  the  rectilinear  course  it  tends  to 
pursue  in  virtue  of  its  inertia.  We  may  then  consider  this  mo- 
tion in  a  curve  line  to  arise  from  two  forces :  one  a  primitive  im- 


CENTRAL  FORCES.  129 

pulse  given  to  the  body,  which  alone  would  have  caused  it  to  de- 
scribe a  straight  line ;  the  other  a  deflecting  force,  which  continu- 
ally urges  the  body  toward  some  point  out  of  the  original  line  of 
motion. 

243.  Kepler's  laws. — Before  Newton's  discovery  of  the  law  of 
universal  gravitation,  the  paths  in  which  the  planets  revolve  about 
the  sun  had  been  ascertained  by  observation ;  and  the  following 
laws,  discovered  by  Kepler,  and  afterward  called  Kepler's  laws. 
were  known  to  be  true : 

1st.  The  radius  vector  of  every  planet  describes  about  the  sun  equal 
areas  in  equal  times. 

2d.  The  path  of  every  planet  is  an  ellipse,  having  the  sun  in  one  of 
its  foci. 

3d.  The  squares  of  the  times  of  revolution  are  as  the  cubes  of  the 
mean  distances  from  the  sun,  or  as  the  cubes  of  the  major  axes  of  tlie 
orbits. 

From  these  facts,  revealed  by  observation,  we  may  deduce  the 
]aw  of  attractive  force  upon  which  they  depend. 

2-1-1.  Theorem. —  When  a  body  moves  in  a  curve,  acted  on  by  a 
force  tending  to  a  fixed  point,  the  areas  which  it  describes  by  radii 
drawn  to  the  centre  of  force  are  in  a  constant  plane,  and  are  propor- 
tional to  the  times. 

Let  S  be  the  centre  of 
attraction;  let  the  time  be 
divided  into  short  and 
equal  portions,  and  in  the 
first  portion  let  the  body 
describe  AB.  In  the  sec- 
ond portion  of  time,  if  no 
new  force  were  to  act  upon 
the  body,  it  would  proceed  S 
to  c  in  the  same  straight 
line,  describing  Be  equal 
to  AB.  But  when  the 
body  has  arrived  at  B,  let 
a  force  tending  to  the  cen- 
tre S  act  on  it  by  a  single 
instantaneous  impulse^and 


130  ASTRONOMY. 

compel  the  body  to  continue  its  motion  along  the  line  BC.  Draw 
Cc  parallel  to  BS,  and  at  the  end  of  the  second  portion  of  time, 
the  body  will  be  found  in  C,  in  the  same  plane  with  the  triangle 
ASB.  Join  SC ;  and  because  SB  and  Cc  are  parallel,  the  triangle 
SBC  will  be  equal  to  the  triangle  SBc,  and  therefore  also  to  the 
triangle  SAB,  because  Be  is  equal  to  BA. 

In  like  manner,  if  a  centripetal  force  toward  S  act  impulsively 
at  C,  D,  E,  etc.,  at  the  end  of  equal  successive  portions  of  time, 
causing  the  body  to  describe  the  straight  lines  CD,  DE,  EF,  etc., 
these  lines  will  all  lie  in  the  same  plane,  and  the  triangles  SCD, 
SDE,  SEF  will  each  be  equal  to  SAB  and  SBC.  Therefore  these 
triangles  will  be  described  in  equal  times,  and  will  be  in  a  con 
stant  plane ;  and  we  shall  have 

polygon  SADS :  polygon  SAFS : :  time  in  AD  :  time  in  AF. 

Let  now  the  number  of  the  portions  of  time  in  AD,  AF  be  aug- 
mented, and  their  magnitude  be  diminished  in  infinitum,  the  pe- 
rimeter ABCDEF  ultimately  becomes  a  curve  line,  and  the  force 
which  acted  impulsively  at  B,  C,  D,  E,  etc.,  becomes  a  force  which 
acts  continually  at  all  points.    Therefore,  in  this  case  also,  we  have 
curvilinear  area  SADS :  curvilinear  area  SAFS 
: :  time  in  AD  :  time  in  AF. 

245.  Theorem. — The  velocity  of  a  body  moving  in  a  curve  and  at- 
tracted to  a  faced  centre,  is  inversely  as  the  perpendicular  from  the 
fixed  centre  upon  the  tangent  to  the  curve. 

For  the  velocities  in  the  polygon  at  two  points,  A,  E,  are  as 
AB,  EF,  because  these  lines  are  described  in  equal  portions  of 
time.  But  if  SY,  SZ  be  drawn  perpendicular  to  these  lines. 
SY.AB  =  SZ.EF,  because  the  triangles  SAB,  SEF  are  equal 
Therefore  velocity  at  A  :  velocity  at  E  : :  SZ  :  SY. 

And  ultimate!}7',  the  velocity  in  the  polygon  becomes  the  veloc- 
ity in  the  curve,  and  the  lines  AY,  EZ  are  the  tangents  to  the 
curve  at  A  and  E. 

246.  Theorem. — If  a  body  moves  in  a  curve  line  in  a  constant 
plane,  and  by  a  radius  drawn  to  a  fixed  point,  describes  areas  about 
tiiat  point  proportional  to  tiie  times,  it  is  urged  by  a  central  force  tend- 
ing to  tiiat  point. 

Every  body  which  moves  in  a  curve  line  is  deflected  from  a 
straight  line  by  some  force  acting  upon  it.  If  the  body  were  to 


CENTRAL   FORCES. 


131 


describe  the  polygon  ABCDEF,  describing  the  equal  triangles 
SAB,  SBC,  etc.,  in  equal  times,  it  must  at  B  be  acted  on  by  a  force 
directed  toward  S.  For  in  AB  produced,  take  Be  equal  to  AB. 
Then  the  triangle  ASB  =  BSc.  But,  by  supposition,  ASB = BSC. 
Therefore,  BSC = BSc;  and,  consequently,  Cc  is  parallel  to  SB. 
Now  BG  may  be  regarded  as  the  resultant  of  two  forces,  one  the 
impulse  in  the  direction  of  AB  produced,  and  the  other  a  deflect- 
ing force  Cc,  which  is  parallel  to  SB ;  that  is,  the  deflecting  force 
at  B  is  directed  toward  the  sun.  But  ultimately  the  motion  in 
the  polygon  will  coincide  with  the  motion  in  the  curve,  and  the 
force  in  the  polygon  will  be  the  same  as  the  force  in  the  curve. 
Therefore  in  the  curvilinear  motion  the  proposition  is  true. 

Now  since  the  planets  describe  about  the  sun  equal  areas  in 
equal  times,  it  follows  that  the  force  which  deflects  them  from  a 
straight  line  is  directed  toward  the  centre  of  the  sun. 

247.  Theorem.  —  When  bodies  describe  different  circles  with  uni- 
form motions,  the  forces  tend  to  the  centres  of  the  circles,  and  are  as  the 
squares  of  the  velocities  divided  by  the  radii  of  the  circles. 

By  Art.  246  the  forces  tend  to 
the  centres  of  the  circles.  Let 
AC,  ac  be  arcs  described  in  two 
different  circles  in  equal  times. 
Draw  the  tangents  AB,  ab ; 
draw  BC,  be  perpendicular  to 
the  tangents,  and  CD,  cd  paral- 
lel to  them.  Draw  also  the 
chords  AC,  ac.  Then  BC,  be,  or  AD,  ad,  are  the  spaces  through 
which  the  bodies  are  deflected  from  the  tangents  by  the  action 
of  the  forces  to  S  and  s.  Then 


Also 


Fig.  6S. 


/ 
AD  :  AC  ::  AC  :  AE  ;  whence  AD  =  ^. 


ac2 
—  . 
2as 


Now  when  the  arc  is  taken  indefinitely  small,  we  shall  have 
the  centripetal  force  at  A  :  centripetal  force  at  a : :  the  square  of 
the  arc  AC  divided  by  the  radius  AS :  the  square  of  the  arc  ac 
divided  by  the  radius  as.  But  the  arcs  AC,  ac,  described  in  equal 
times,  are  as  the  velocities ;  hence  in  circles,  if  F  represent  the 
centripetal  force,  V  the  velocity,  and  E  the  radius  of  the  circle. 


132 


ASTRONOMY. 


we  shall  have 


or 


V2 

F  varies  as-rr. 

XV 


248.  Theorem.  —  When  bodies  describe  different  circles  with  uniform 
motions  •,  *Ae  central  force  is  as  the  radius  of  the  circle  divided  by  the 
square  of  the  time  of  one  revolution. 

Let  E  be  the  radius  of  the  circle,  V  the  velocity,  and  T  the  time 
of  describing  the  whole  circle.  The  circumference  of  the  circle 

will  be  represented  by  2?rR,  which  equals  VT  ;  hence  ~¥= 


—r-r. 

V2       4;r2E2  E 

But  by  Art.  247,  F  varies  as  -,  or  ,  which  varies  as  r—  ; 


that  is. 


Fig.  CO. 


249.  Theorem. — If  a  body  describes  an  ellipse,  being  continually 
urged  by  a  fcrce  directed  toward  the  focus,  that  force  must  vary  in- 
versely as  the  square  of  the  distance. 

Let  APB  represent 
the  elliptic  orbit  of  a 
planet,  and  S  the  fo- 
cus occupied  by  the 
sun.  Let  PQ  be  an 
arc  described  by  the 
planet  in  an  indefi- 
nitely short  time,  t. 
Draw  the  diameter 
PG;  also  the  ordinate 
Qy  parallel  to  the  tan- 
gentatP;  and  let  DK 
be  the  diameter  which  is  conjugate  to  PG.  Draw  the  radius  vec- 
tor SP,  cutting  the  diameter  DK  in  E,  and  the  ordinate  Qv  in  x, 
and  complete  the  parallelogram  QxPE.  Also  draw  QT  perpen- 
dicular to  SP,  and  PF  perpendicular  to  DK. 

If  the  arc  PQ  be  taken  indefinitely  small,  it  may  be  considered 
as  a  straight  line  described  by  the  joint  action  of  the  force  which 
is  directed  toward  S,  and  of  the  projectile  force  which  acts  in  the 
direction  of  PR  That  is,  the  force  PQ  may  be  resolved  into  the 


CENTRAL   FORCES.  133 

two  forces  Qx  and  Pz.  During  the  time  £,  the  deflecting  force, 
if  it  acted  alone,  would  cause  the  body  to  describe  Px.  Hence, 
denoting  the  intensity  of  this  force  by  F,  we  have,  by  Mechanics, 

Px=$¥l2; 
and  taking  t  for  the  unit  of  time,  we  have 

F  =  2Pz. 

By  similar  triangles,        P*  :  Pi-  :  :  PE  :  PC. 
By  Ellipse,  Prop.  XXL,  Gu.Pv:  Qv*  :  :  PC2  :  CD2. 
Compounding  these  two  proportions,  we  have 
Gy.P;r:Qy2::PC.AC:CD2, 
since  PE  =  AC,  Geom.,  Ellipse,  Prop.  VIII. 

But  \vhen  the  arc  PQ  is  taken  indefinitely  small,  Qy=Qr,  and 
Gy=2PC. 

^  .     Gy.Pz.CD2    2Pz.CD2  ,1N 

Hence  Q,2  =  _____=____.  (l) 

Again,  by  similar  triangles, 

Q*:QT::PE(=CA):PF. 
Also  (Ellipse,  Prop.  XVIII.), 

CD  .  PF=CA  .  CB,  or  CA  :  PF  :  :  CD  :  CB. 
Hence  Qx:  QT:  :  CD:  CB; 

n      QT.CD  QT2.CD2 

.'.Qg=     CB     ,     and  Qx*=      ^     .  (2) 

From  equations  (1)  and  (2)  we  have 

2P;r.CD2_QT2.CD2 
AC  CB2     ' 

Represent  half  the  major  axis  of  the  ellipse  by  a,  and  half  the 
minor  axis  by  b  ;  then 

QT2.AC_QT2.q 
CB2  tf     • 

If  now  we  denote  the  area  of  the  elliptical  sector  SQP  by  &, 
we  have  A=|SP.QT; 

97-  d.Z-2 

and  hence  QT=|p  and  QT2-|^. 

Substituting  this  value  in  equation  (3),  we  have 


If  we  consider  the  action  of  the  deflecting  force  at  some  other 
point  of  the  ellipse,  as  P',  and  denote  the  intensity  of  the  force  by 

IHT  -™     4&2     a 

F  ,  we  shall  have  F  =ep^  •  75- 


134  ASTRONOMY. 

But  by  Kepler's  first  law,  k  is  a  constant  quantity ;  hence  we 
have 

F:F'::SP'2:SP2; 

or  the  deflecting  force  varies  inversely  as  the  square  of  the  dis- 
tance of  the  planet  from  the  sun. 

250.  Theorem. —  Wlien  several  bodies  revolve  in  ellipses  about  the 
same  centre  of  force,  varying  inversely  as  the  square  of  the  distance, 
the  squares  of  the  periodic  times  ivill  vary  as  the  cubes  of  the  major 
axes. 

Let  T  denote  the  periodic  time  of  a  planet,  expressed  in  sec- 
onds, and  k  the  area  described  by  the  radius  vector  in  one  second ; 
then  the  entire  area  of  the  ellipse  will  be  represented  by  T&.  But 
this  area  is  also  represented  by  nab  (Ellipse,  Prop.  XXIII.). 

Hence  T^iraft,  or  &=^. 

Eepresent  the  distance  of  the  planet  from  the  sun  by  E ;  then, 
by  the  last  article, 

Tj1 

If  we  represent  by  /  the  value  of  the  deflecting  force  F  at  the 
distance  of  unity,  then,  by  hypothesis, 

/:F::p:i;  that  is,  F  =  ^. 

f  A       o  .  1 

Hence  — - 


that  is 


If  T'  denote  the  periodic  time  of  a  second  planet,  and  a!  half 
the  major  axis  of  its  orbit,  we  shall  have 


,  r  Vf  ' 

whence  T  :  T'  :  :  a*  :  a*,  or  T2  :  T2  :  :  a3  :  a'3. 

Thus  we  perceive  that  Kepler's  first  law  would  hold  true,  what- 
ever might  be  the  law  by  which  the  deflecting  force  depended 
upon  the  distance  ;  but  the  second  and  third  laws  prove  that  in 
the  solar  system  this  deflecting  force  varies  inversely  as  the  square 
of  the  distance. 


LAW   OF   GRAVITATION.  135 

251.  Modification  of  Kepler's  third  laic. — Kepler's  third  law  is 
strictly  true  only  in  the  case  of  planets  whose  quantity  of  matter 

,  is  inappreciable  in  comparison  with  that  of  the  central  body.  In 
considering  the  motion  of  a  planet,  for  instance  Jupiter,  round  the 
sun,  it  is  necessary  to  remember  that  while  the  sun  attracts  Ju- 
piter, Jupiter  also  attracts  the  sun.  The  motion  which  the  attrac- 
tion of  Jupiter  produces  in  the  sun,  is  less  than  the  motion  which 
the  attraction  of  the  sun  produces  in  Jupiter,  in  the  same  ratio  in 
which  Jupiter  is  smaller  than  the  sun.  If  the  sun  and  Jupiter 
were  allowed  to  approach  one  another,  their  rate  of  approach 
would  be  the  sum  of  the  motions  of  the  sun  and  Jupiter,  and 
would  therefore  be  greater  than  their  rate  of  approach  if  the  sun 
were  not  movable,  in  the  same  ratio  in  which  the  sum  of  the 
masses  of  the  sun  and  Jupiter  is  greater  than  the  sun's  mass. 
Consequently,  in  comparing  the  orbits  described  by  different  plan- 
ets round  the  sun,  we  must  suppose  the  central  force  to  be  the  at- 
traction of  a  mass  equal  to  the  sum  of  the  sun  and  planet 

If  we  regard  the  mass  of  the  sun  as  unity,  and  represent  the 
masses  of  two  planets  by  m  and  m',  then  we  shall  have 

T2 .  T/2 . .     ft3     .     a'3    . 
"  I  +  m  '  1  +  m' ' 
and  this  proportion  is  rigorously  true. 

252.  The  force  that  retains  the  moon  in  her  orbit  is  the  same  as 
that  which  causes  bodies  to  fall  near  the  eartJCs  surface,  the  force  being 
diminished  in  proportion  to  the  square  of  the  distance  from  the  eartits 
centre. 

Let  E  be  the  centre  of  the  earth,  A  a  point 
on  its  surface,  and  BC  a  part  of  the  moon's  or-  D_ 
bit  assumed  to  be  circular.  When  the  moon  is 
at  any  point,  B,  in  her  orbit,  she  would  move  on 
in  the  direction  of  the  line  BD,  a  tangent  to  the 
orbit  at  B,  if  she  was  not  acted  upon  by  some 
deflecting  force.  Let  F  be  her  place  in  her  orbit 
one  second  of  time  after  she  was  at  B,  and  let 
FGr  be  drawn  parallel  to  BD,  and  FH  parallel 
to  EB.  The  line  FH,  or  its  equal  BG-,  is  the 
distance  the  moon  has  been  drawn,  during  one 
second,  from  the  tangent  toward  the  earth  at  E. 
If  we  divide  the  circumference  of  the  moon's 


13tf  ASTRONOMY. 

orbit  by  the  number  of  seconds  in  the  time  of  one  revolution,  we 
shall  have  the  length  of  the  arc  BF.     Now,  by  Geometry, 
2BE:BF::BF:BG. 

But  the  chord  BF  does  not  differ  sensibly  from  the  arc  BF, 
already  obtained.  BG  is  thus  found,  by  computation,  to  be  0.0534 
inch. 

At  the  equator,  a  body  falls  through  192£  inches  in  the  first 
second.  At  the  distance  of  the  moon,  the  force  of  gravity  (if  it 
diminishes  in  proportion  to  the  square  of  the  distance  from  the 
earth's  centre)  will  be  found  by  the  proportion 

S9.9642 : 12 : :  192£ :  0.0535  inch, 
which  agrees  very  nearly  with  the  distance  above  computed. 

The  space  through  which  the  moon  actually  falls  toward  the 
earth  in  one  second  is  a  little  less  than  that  computed  from  the 
force  of  gravity  at  the  earth's  surface,  because  (as  we  shall  see 
hereafter)  the  action  of  the  sun  diminishes  by  a  small  quantity 
the  moon's  gravity  toward  the  earth. 

253.  Deductions  from  Kepler 's  laivs. — By  Kepler's  first  law,  the 
radius  vector  of  each  planet  describes  about  the  sun  equal  areas 
in  equal  times;  hence  it  follows  (Art.  246)  that  each  planet  is 
acted  upon  by  a  force  which  urges  it  continually  toward  the  cen- 
tre of  the  sun.  We  say,  therefore,  that  the  planets  gravitate  to- 
ward the  sun,  and  the  force  which  urges  each  planet  toward  the 
sun  is  called  its  gravity  toward  the  sun.  By  Kepler's  second  law, 
the  planets  describe  ellipses,  having  the  centre  of  the  sun  at  one 
of  their  foci ;  hence  it  follows  (Art  249)  that  the  force  of  gravity 
of  each  planet  toward  the  sun  varies  inversely  as  the  square  of  its 
distance  from  the  sun's  centre.  By  Kepler's  third  law,  the  squares 
of  the  times  of  revolution  of  the  different  planets  are  as  the  cubes 
of  the  mean  distances  from  the  sun ;  hence  it  follows  (Art.  250) 
that  the  planets  are  solicited  by  a  force  of  gravitation  toward  the 
sun,  which  varies  from  one  planet  to  another  inversely  as  the 
square  of  their  distance.  It  is,  therefore,  the  same  force,  modified 
only  by  distance  from  the  sun,  which  causes  all  the  planets  to 
gravitate  toward  him,  and  retains  them  in  their  orbits.  This 
force  is  conceived  to  be  an  attraction  of  the  matter  of  the  sun  for 
the  matter  of  the  planets,  and  is  called  the  solar  attraction.  This 
force  extends  infinitely  in  every  direction,  varying  inversely  as 
the  square  of  the  distance. 


LAW   OF   GRAVITATION.  137 

254.  The  planets  endowed  with  an  attractive  force. — The  motions 
of  the  satellites  about  their  primary  planets  are  also  found  to  be 
in  conformity  with  Kepler's  laws;  hence  we  conclude  that  the 
planets  which  have  satellites  are  endowed  with  an  attractive  force, 
which  extends  indefinitely  in  every  direction,  and  varies  inverse- 
ly as  the  square  of  the  distance.     It  is  evident,  also,  that  the  sat- 
ellites gravitate  toward  the  sun  in  the  same  manner  as  their 
planets,  for  their  relative  motions  about  their  primaries  are  the 
same  as  if  the  planets  were  at  rest. 

The  planets  which  have  no  satellites  are  endowed  with  a  sim- 
ilar attractive  force,  as  is  proved  by  the  disturbances  which  they 
cause  in  the  motion  of  the  other  planets. 

255.  The  component  particles  of  the  sun  and  planets  attract  each 
other. — The  force  of  attraction  of  one  body  for  another  arises  from 
the  attraction  of  its  component  particles.     A  large  planet  may  be 
regarded  as  a  collection  of  numerous  smaller  planets,  and  the  at- 
traction of  the  whole  must  be  the  result  of  the  attraction  of  the 
component  parts.     Thus  the  gravitation  of  the  earth  toward  the 
sun  is  the  sum  of  the  gravitation  of  its  component  particles,  and 
thus,  also,  the  force  of  gravity  of  each  of  the  planets  is  propor- 
tional to  the  matter  which  it  contains ;  that  is,  to  its  mass.     More- 
over, since  the  attraction  of  the  planets  varies  inversely  as  the 
square  of  the  distance,  the  force  of  every  particle  must  also  vary 
inversely  as  the  square  of  the  distance  of  the  particles. 

256.  Theory  of  universal  gravitation. — It  follows,  then,  as  a  nec- 
essary consequence,  from  the  general  facts  or  laws  discovered  by 
Kepler,  that  all  bodies  mutually  attract  each  other,  with  forces 
varying  directly  as  their  quantities  of  matter,  and  inversely  as 
the  squares  of  their  distances.     This  principle  is  called  the  law  of 
universal  gravitation.     It  was  first  distinctly  promulgated  by  Sir 
Isaac  Newton,  and  hence  is  frequently  called  Newton's  Theory  of 
Universal  Gravitation.     If  we  represent  the  mass  of  the  sun  by 
M,  and  the  distance  of  a  planet  from  the  sun  by  E,  then  the  at- 
traction of  the  sun  upon  the  planet  will  be  represented  by  ^. 

A  similar  formula  will  represent  the  attraction  of  a  planet  upon 
its  satellite. 

This  universal  gravitation  disturbs  the  motion  of  the  planets 


138  ASTRONOMY". 

in  their  orbits  about  the  sun.  If  the  earth  were  attracted  only 
by  the  sun,  it  would  describe  an  exact  ellipse ;  but,  since  it  is  also 
attracted  by  the  other  planets,  it  is  continually  drawn  out  of  the 
orbit  in  which  it  had  previously  been  moving.  The  satellites 
also,  being  disturbed  in  their  motions  about  their  planets  by  their 
mutual  attractions  and  by  that  of  the  sun,  are  continually  drawn 
out  of  the  orbits  in  which  they  were  previously  moving. 

257.  The  heavenly  bodies  all  move  in  conic  sections. — It  was  de- 
monstrated by  Newton  that  if  a  body  (a  planet,  for  instance)  is 
impelled  by  a  projectile  force,  and  is  continually  attracted  toward 
the  sun's  centre  by  a  force  varying  inversely  as  the  square  of  the 
distance,  and  no  other  forces  act  upon  the  body,  the  body  will 
move  in  one  of  the  following  curves — a  circle,  an  ellipse,  a  para- 
bola, or  an  hyperbola ;  that  is,  it  will  move  in  one  of  the  conic 
sections.  The  form  of  the  orbit  will  depend  upon  the  direction 
and  intensity  of  the  projectile  force. 

If  we  conceive  F  to  be 
the  centre  of  an  attractive 
force,  and  a  body  at  A  to  be 
projected  in  a  direction  at 
right  angles  to  the  line  AF, 
then  there  is  a  certain  ve- 
locity of  projection  which 
would  cause  the  body  to 
describe  the  circle  ABC; 
a  greater  velocity  would 
cause  it  to  describe  the  el- 
lipse ADE,  or  the  more  ec- 
centric ellipse  AGH;  and 
if  the  velocity  of  projection  be  sufficient,  the  body  will  describe 
the  semi-parabola  AKL.  If  the  velocity  of  projection  be  still 
greater,  the  body  will  describe  an  hyperbola.  The  curve  can  not 
be  a  circle  unless  the  body  be  projected  in  a  direction  perpendicu- 
lar to  AF,  and,  moreover,  unless  the  velocity  with  which  the 
planet  is  projected  is  neither  greater  nor  less  than  one  particular 
velocity,  determined  by  the  length  of  FA  and  the  mass  of  the  cen- 
.1  body.  If  it  differs  little  from  this  particular  velocity  (either 
reater  or  less),  the  body  will  move  in  an  ellipse;  but  if  it  is 
much  greater,  the  body  will  move  in  a  parabola  or  an  hyperbola. 


MOTIONS  OF  PROJECTILES. 


139 


If  the  body  be  projected  in  a  direction  AB        Fig.  72. 
oblique  to  SA,  and  the  velocity  of  projection  is 
small,  the  body  will  move  in  an  ellipse ;  but  if 
the  velocity  is  great,  it  may  move  in  a  parabola 
or  hyperbola,  but  not  in  a  circle. 

If  a  body  describe  a  circle,  the  sun  is  in  the 
centre  of  the  circle.  If  the  body  describe  an  el- 
lipse, the  sun  is  not  in  the  centre  of  the  ellipse, 
but  in  one  focus.  If  the  body  describe  a  para- 
bola or  an  hyperbola,  the  sun  is  in  the  focus.  The  planets  de- 
scribe ellipses  which  differ  little  from  circles.  A  few  of  the  com- 
ets describe  very  long  ellipses ;  and  nearly  all  the  others  that  have 
been  observed  are  found  to  move  in  curves  which  can  not  be  dis- 
tinguished from  parabolas.  There  is  reason  to  think  that  two  or 
three  comets  which  have  been  observed  move  in  hyperbolas. 

258.  Motions  of  projectiles. — The  motions  of  projectiles  are  gov- 
erned by  the  same  laws  as  the  motions  of  the  planets.  If  a  body 
be  projected  in  a  horizontal  direction  from  the  top  of  a  mountain, 
it  is  deflected  by  the  attraction  of  the  earth  from  the  rectilinear 
path  which  it  would  otherwise  have  pursued,  and  made  to  de- 
scribe a  curve  line  which  at  length  brings  it  to  the  earth's  sur- 
face; and  the  greater  the  velocity  of  projection,  the  farther  it  will 
go  before  it  reaches  the  earth's  surface.  We  may  therefore  sup- 
pose the  velocity  to  be  so  increased  that  it  shall  pass  entirely 
round  the  earth  without  touching  it. 

Let  BCD  represent  the  sur-  Fi(I  73 

face  of  the  earth;  AB,  AC, 
AD  the  curve  lines  which  a 
body  would  describe  if  project- 
ed horizontally  from  the  top  of 
a  high  mountain,  with  success- 
ively greater  and  greater  ve-  H| 
locities.  Supposing  there  were 
no  air  to  offer  resistance,  and 
the  velocity  were  sufficiently 
great,  the  body  would  pass  en- 
tirely round  the  earth,  and  re- 
turn to  the  point  from  which 
it  was  projected. 


140  ASTRONOMY. 

259.  Ti me  of  revolution  near  the  earth's  surface. — By  means  of 
Kepler's  third  law,  we  are  able  to  compute  the  time  required  to 
complete  a  revolution  in  such  an  orbit  near  the  earth's  surface. 
We  may  regard  such  a  body  as  a  satellite  revolving  round  the 
earth's  centre  in  an  orbit  whose  radius  is  equal  to  the  radius  of 
the  earth,  while  the  moon  completes  one  revolution  in  27.32  da}rs 
in  an  orbit  whose  radius  is  59.96  times  the  radius  of  the  earth. 
If  we  put  T  to  represent  the  periodic  time  of  such  a  satellite,  we 
shall  have  the  proportion 

59.963:13::27.322:T2; 
from  which  we  find  T= 0.0588  day,  or  Ih.  24m.  35s. 

If  the  velocity  of  projection  were  too  small  to  carry  it  entirely 
round  the  earth,  and  the  impenetrability  of  the  earth  did  not  pre- 
vent, it  would  describe  an  ellipse,  of  which  the  earth's  centre 
would  occupy  the  lower  focus,  and  it  would  return  again  to  the 
point  from  which  it  started.  This  conclusion  is  easily  reconciled 
with  the  doctrine  of  Mechanics,  that  the  path  of  a  projectile  is  a 
parabola,  for  it  is  there  assumed  that  gravity  acts  in  parallel  di- 
rections, and  that  it  is  a  constant  accelerating  force.  These  prin- 
ciples are  sensibly  true  for  small  distances,  but  they  are  not  true 
when  great  distances  are  considered. 

Problem. — How  much  faster  than  at  present  must  the  earth  ro- 
tate upon  its  axis,  in  order  that  bodies  on  its  surface  at  the  equa- 
tor may  lose  all  their  gravity  ?  Ans.  17  times. 

260.  Why  a  planet  at  perihelion  does  not  fall  to  the  sun. — Since 
the  sun's  force  of  attraction  is  greatest  when  the  distance  is  least, 
it  might  seem  that  when  a  planet  has  reached  its  perihelion  it 
must  inevitably  fall  to  the  sun.     The  planet,  however,  recedes 
from  the  sun,  partly  on  account  of  the  increased  velocity  near 
perihelion,  and  partly  on  account  of  the  gradual  change  in  its  di- 
rection.    The  curvature  of  any  part  of  a  planetary  orbit  depends 
not  solely  upon  the  force  of  the  sun's  attraction,  but  also  on  the 
velocity  with  which,  the  planet  is  moving.     The  greater  the  ve- 
locity of  the  planet,  the  less  will  be  the  curvature  of  the  orbit. 

Suppose  a  planet  to  have  passed  the  aphelion  A  with  so  small 
a  velocity  that  the  sun's  attraction  bends  the  path  very  much, 
and  causes  it  immediately  to  begin  to  approach  toward  the  sun; 
the  sun's  attraction  will  increase  its  velocity  as  it  moves  through 
B,  C,  and  D ;  for  when  the  planet  is  at  B;  the  sun's  attractive 


MOTION  IN  AN  OBBIT.  141 

rig.  74    A  f°rce  acts  *n  t^e  Direction  BS ;   and,  on  ac- 

count of  the  small  inclination  of  BC  to  BS,  the 
force  acting  in  the  direction  of  BS  increases 
the  planet's  velocity.  Thus  the  planet's  ve- 
locity is  continually  increasing  as  the  planet 
moves  through  B,  C,  and  D ;  and  although, 
on  account  of  the  planet's  nearness,  the  sun's 
attractive  force  is  very  much  increased,  and 
tends  therefore  to  make  the  orbit  more  curved, 
yet  the  velocity  is  so  much  increased  that  the 
orbit  is  no  more  curved  at  E  than  it  was  at  A;  and  at  perihelion 
the  velocity  is  so  great  that  the  planet  begins  immediately  to  re- 
cede from  the  sun. 

A  similar  course  of  reasoning  will  explain  why,  when  the  plan- 
et reaches  its  greatest  distance  from  the  sun,  where  the  sun's  at- 
traction is  least,  it  does  not  altogether  fly  off  from  the  sun.  As 
the  planet  passes  through  F,  G,  II,  the  sun's  attraction,  which  is 
always  directed  toward  S,  retards  the  planet  in  its  orbit,  and  when 
it  has  reached  A  its  velocity  is  extremely  small ;  and  therefore, 
although  the  sun's  attraction  at  A  is  small,  yet  the  deflection 
which  it  produces  in  the  planet's  motion  is  such  as  to  give  its  path 
the  same  curvature  as  at  E.  Then  the  planet  again  approaches 
the  sun,  and  goes  over  the  same  orbit  as  before. 

261.  Could  the  rotary  and  orbital  motions  of  the  earth  have  been 
caused  by  a  single  force  ? — It  is  possible  that  the  rotary  motion  of 
the  earth,  and  its  motion  in  its  orbit  about  the  sun,  are  both  the 
result  of  a  single  primitive  impulse.  If  a  sphere  were  to  receive 
an  impulse  in  the  direction  of  its  centre  of  gravity,  it  would  have 
a  progressive  motion  without  any  rotation  upon  an  axis.  But  if 
the  impulse  were  given  in  any  other  direction,  it  would  produce 
also  a  rotary  motion.  It  is  possible  to  compute  at  what  distance 
from  the  centre  of  gravity  an  impulse  must  be  given  to  produce 
the  actual  progressive  and  rotary  motions  observed  in  a  body.  In 
order  to  explain  the  motion  of  the  earth  in  its  orbit,  and  that  of 
its  rotation  upon  an  axis  in  24  hours,  the  impulse  must  have  been 
given  in  a  line  passing  24  miles  from  the  centre  of  the  earth. 


1-12  ASTRONOMY. 


262.  PROBLEMS. 

Prob.  1.  The  mean  distance  of  the  planet  Hygeia  from  the  sun 
is  3.14937  (the  distance  of  the  earth  being  taken  as  unity) ;  re- 
quired its  periodic  time  ? 

By  Art.  250,  a3 :  a'3 ::  T2:  T/2; 

that  is,  I3 :  3.149373 : :  3G5.252 :  T2. 

Ans.  2041.4  days. 

Prob.  2.  The  periodic  time  of  the  planet  Flora  is  1193  days ; 
required  its  mean  distance  from  the  sun?  Ans. 2.2013. 

Prob.  3.  What  would  be  the  periodic  time  of  a  planet  revolving 
about  the  sun  at  a  mean  distance  of  ten  million  miles  ? 

Prob.  4.  What  would  be  the  periodic  time  of  a  planet  revolving 
about  the  sun  at  a  mean  distance  of  one  million  miles  ? 

Prob.  5.  Suppose  there  exists  a  planet  revolving  about  the  sun 
at  a  mean  distance  of  5000  millions  of  miles,  what  must  be  its  pe- 
riodic time  ? 

Prob.  6.  What  would  be  the  periodic  time  of  a  satellite  revolv- 
ing about  the  earth  at  a  mean  distance  of  10,000  miles  from  the 
earth's  centre? 

Prob.  7.  Suppose  the  earth  had  a  satellite  making  one  revo- 
lution in  a  year,  what  would  be  its  mean  distance  from  the 
earth? 

'263.  The  probkm  of  the  three  bodies. — When  there  are  only  two 
bodies  that  gravitate  to  one  another  with  forces  inversely  as  the 
squares  of  their  distances,  they  move  in  conic  sections,  and  de- 
scribe about  their  common  centre  of  gravity  equal  areas  in  equal 
times.  But  if  there  are  three  bodies,  the  action  of  any  one  on  the 
other  two,  changes  the  form  of  their  orbits,  so  that  the  determina- 
tion of  their  motions  becomes  a  problem  of  great  difficulty,  distin- 
guished by  the  name  of  the  probkm  of  the  three  bodies. 

The  solution  of  this  problem,  in  its  utmost  generality,  has  never 
been  effected.  Under  certain  limitations,  however,  and  such  as 
are  quite  consistent  with  the  condition  of  the  heavenly  bodies,  it 
admits  of  being  resolved.  The  most  important  of  these  limita- 
tions is  that  the  force  which  one  of  the  bodies  exerts  upon  the 
other  two  is,  either  from  the  smallness  of  that  body  or  its  great 
distance,  very  inconsiderable,  in  respect  of  the  forces  which  these 
two  exert  on  one  another. 


LUNAR  IRREGULARITIES.  14:3 

The  force  of  this  third  body  is  called  a  disturbing  force,  and  its 
effects  in  changing  the  places  of  the  other  two  bodies  are  called 
the  disturbances,  or  perturbations  of  the  system. 

Though  the  small  disturbing  forces  may  be  more  than  one,  or 
though  there  be  a  great  number  of  remote  disturbing  bodies,  their 
combined  effect  may  be  computed,  and  therefore  the  problem  of 
three  bodies,  under  the  conditions  just  stated,  may  be  extended  to 
any  number. 

264.  How  the  moon's  elliptic  motion  is  disturbed. — The  only  body 
in  the  solar  system  which  produces  a  sensible  disturbing  effect 
upon  the  moon  is  the  sun ;  for  although  several  of  the  planets 
sometimes  come  within  less  distances  of  the  earth,  their  masses 
are  too  inconsiderable  to  produce  any  sensible  disturbing  effect 
upon  the  moon's  motion.     The  mass  of  the  sun,  on  the  contrary, 
is  so  great,  that,  although  the  radius  of  the  moon's  orbit  bears  a 
small  ratio  to  the  sun's  distance,  and  although  lines  drawn  from 
the  sun  to  any  part  of  that  orbit  are  nearly  parallel,  the  differ- 
ence between  the  forces  exerted  by  the  sun  upon  the  moon  and 
earth  is  quite  sensible. 

265.  Relative  attractions  of  the  sun  and  earth  upon  the  moon. — It 
was  shown,  Art.  252,  that  the  earth  draws  the  moon  from  a  tan- 
gent 0.0534  inch  in  a  second.     If  a  similar  calculation  be  made 
in  relation  to  the  orbit  of  the  earth,  it  will  be  found  that  the  sun 
draws  the  earth  from  a  tangent  0.119  inch  in  a  second.    Also,  the 
average  force  which  the  sun  exerts  upon  the  moon  must  be  the 
same  as  that  which  it  exerts  upon  the  earth ;  that  is,  the  sun  ex- 
erts  upon  the  moon  a  force  2-g-  times  as  great  as  the  earth  does. 
The  moon  is  therefore  much  more  under  the  influence  of  the  sun 
than  of  the  earth. 

266.  Mass  of  the  sun  compared  with  that  of  the  earth. — The  force 
which  the  sun  exerts  on  the  earth  is  2-g-  times  greater  than  that 
which  the  earth  exerts  on  the  rnoon.     But  the  force  of  attraction 
varies  inversely  as  the  square  of  the  distance,  and  the  distance  of 
the  sun  from  the  earth  is  about  400  times  the  distance  of  the 
moon.     Hence,  if  the  sun  were  at  the  same  distance  as  the  moon, 
his  force  of  attraction  would  be  the  square  of  400,  or  160,000 
times  as  great  as  it  is  now;  that  is,  it  would  be  2-^x  160,000,  or 


144  ASTRONOMY. 

352,000  times  as  great  as  the  earth's  attraction,  and,  consequently, 
must  have  352,000  times  as  much  matter. 

The  best  determination  of  the  sun's  mass  is  considered  to  be 
354,936. 

267.  How  the  sun's  attraction  acts  as  a  disturbing  force. — If  the 
sun  were  at  an  infinite  distance,  the  earth  and  moon  would  be  at- 
tracted equally  and  in  parallel  straight  lines,  and,  in  that  case, 
their  relative  motions  would  not  be  in  the  least  disturbed.     But 
although  the  distance  of  the  sun  compared  with  that  of  the  moon 
is  very  great,  it  can  not  be  considered  infinite.     The  moon  is  al- 
ternately nearer  to  the  sun  and  farther  from  him  than  the  earth, 
and  the  straight  line  which  joins  her  centre  and  that  of  the  sun 
forms  with  the  terrestrial  radius  vector  an  angle  which  is  contin- 
ually varying.     Thus  the  sun  acts  unequally  and  in  different  di- 
rections on  the  earth  and  moon,  and  hence  result  inequalities  in 
her  motion,  which  depend  on  her  position  in  respect  of  the  sun. 

268.  General  effect  of  the  sun's  disturbing  action. — Let  us  suppose 
that  the  projectile  motions  of  the  earth  and  moon  are  destroyed, 
and  that  they  are  allowed  to  fall  freely  toward  the  sun.     If  the 
moon  was  in  conjunction  with  the  sun,  it  would  be  more  attract- 
ed than  the  earth,  and  fall  with  greater  velocity  toward  the  sun, 
so  that  the  distance  of  the  moon  from  the  earth  would  be  increased 
in  the  fall.     If  the  moon  was  in  opposition,  she  would  be  less  at- 
tracted than  the  earth  by  the  sun,  and  would  fall  with  a  less  ve- 
locity toward  the  sun  than  the  earth,  and  the  moon  would  be  left 
behind  by  the  earth,  so  that  the  distance  of  the  moon  from  the 
earth  would  be  increased  in  this  case  also.     If  the  moon  was  in 
one  of  the  quarters,  then  the  earth  and  moon,  being  both  attracted 
toward  the  centre  of  the  sun,  would  approach  the  sun,  and  at  the 
same  time  would  necessarily  approach  each  other,  so  that  their 
distance  from  each  other  would  in  this  case  be  diminished.    Now 
whenever  the  action  of  the  sun  would  increase  their  distance  if 
they  were  allowed  to  fall  toward  the  sun,  it  produces  the  same 
effect  as  if  their  gravity  to  each  other  was  diminished ;  and  when- 
ever the  action  of  the  sun  would  diminish  their  distance,  their 
gravity  to  each  other  is  increased.     Hence  we  conclude  that  the 
sun's  action  increases  the  gravity  of  the  moon  to  the  earth  at  the  quad- 
ratures, and  diminishes  it  at  the  syzygies. 


LUNAR  IRREGULARITIES. 


145 


Fig.  75. 


269,  How  to  estimate  the  amount  of  the  sun's  disturbing  force. — 
We  may  estimate  the  amount  of  tins  disturbing  force  in  the  fol- 
lowing manner : 

Let  ABCD  represent  the  orbit  of  the  moon, 
with  the  earth  at  E,  and  let  the  sun  be  at  S 
and  the  moon  at  M.    Let  the  line  SE  be  taken 
|B   to  represent  the  force  with  which  the  sun  at- 
tracts the  earth ;  then  we  may  determine  the 
magnitude  of  the  force  with  which  the  sun 
acts  on  the  moon  at  M  by  the  proportion 
SE3 


SM2:SE2::SE: 


duced 


JSM2 


In  the  line  MS,  pro- 


if  necessary,  take  MG=-T-TT^  and  it 

OAL 

will  represent  the  force  with  which  the  sun 
attracts  the  moon.  We  may  suppose  the 
force  MG  to  result  from  the  combined  action 
of  two  forces,  MF  and  MH  (MG  being  the 
diagonal  of  the  parallelogram  MFGH),  of 
which  one,  MF,  is  equal  and  parallel  to  ES. 
Now  if  the  earth  and  moon  were  only  acted 
upon  by  the  equal  and  parallel  forces  ES  and 
n  MF,  their  relative  motions  would  not  be  af- 
fected. Therefore  it  is  only  MH  which  dis- 
turbs this  relative  motion  ;  that  is,  MH  rep- 
resents the  quantity  and  direction  of  this  dis- 
turbing force.  This  force,  MH,  may  be  resolved  into  two  forces, 
MK,  ML,  the  first  being  in  the  direction  of  the  radius  vector  ME, 
and  the  other  having  the  direction  of  a  tangent  to  the  orbit.  The 
force  MK  augments  or  diminishes  the  moon's  gravitation  to  the 
earth  ;  while  the  force  ML  affects  the  moon's  angular  motion 
round  the  earth,  sometimes  accelerating  and  sometimes  retard- 
ing it. 

It  is  evident  that  the  tangential  force  LM  retards  the  moon's 
motion  when  going  from  A  to  B.  If  we  construct  a  similar  fig- 
ure for  each  of  the  other  quadrants,  we  shall  find  that  the  tangen- 
tial force  accelerates  the  moon's  motion  from  D  to  A,  and  also  from 
B  to  C,  but  retards  the  moon's  motion  when  going  from  C  to  D. 
This  force  becomes  zero  at  each  of  the  points  A,  B,  C,  and  D,  and 
has  its  maximum  value  near  the  octants. 


146  ASTRONOMY. 

"When  the  moon  is  in  conjunction,  the  disturbing  force  of  the 
sun  is  wholly  employed  in  drawing  the  moon  away  from  the 
earth  ;  that  is,  in  diminishing  the  moon's  gravitation  to  the  earth. 
When  the  moon  is  in  opposition,  the  force  with  which  the  sun 
draws  the  earth  is  greater  than  that  with  which  it  draws  the  moon, 
so  that  the  effect  of  the  sun's  attraction  is  to  increase  the  distance 
of  the  moon  from  the  earth ;  that  is,  it  is  the  same  as  if  the  sun's 
force  drew  the  moon  away  from  the  earth,  or  diminished  the 
moon's  gravitation  to  the  earth. 

When  the  moon  is  in  quadrature,  the  tangential  force  disap- 
pears, and  the  disturbing  force  is  wholly  employed  in  augmenting 
the  moon's  gravitation  to  the  earth.  The  sun  attracts  the  earth 
and  moon  equally,  but  not  in  parallel  lines.  If  we  suppose  the 
projectile  motions  of  the  earth  and  moon  to  be  destroyed,  and 
that  they  are  allowed  to  fall  freely  toward  the  sun,  the  earth  and 
moon,  both  moving  toward  the  centre  of  the  sun,  would  approach 
each  other,  and  in  one  second  (their  distance  from  the  sun  being 
400  times  the  radius  of  the  moon's  orbit)  their  distance  from 
each  other  would  be  diminished  by  T^th  part  of  the  space  fallen 
through.  Hence,  if  ES  represents  the  force  of  the  moon's  gravi- 
tation to  the  sun,  then  BE  will  represent  the  augmentation  of  the 
moon's  gravitation  to  the  earth  in  quadratures. 

-270.  Numerical  estimate  of  the  suns  disturbing  force. — The  ratio 
of  the  line  ME!  to  ES  may  be  computed  by  Trigonometry  when 
we  know  the  distance  of  the  sun  and  moon  from  the  earth,  and 
also  the  angular  distance  of  the  moon  from  the  sun.  Also  the 
disturbing  force  of  the  sun  upon  the  moon  may  be  compared  with 
the  earth's  attraction  upon  the  moon  by  the  following  propor- 
tions: 

1st.  Disturbing  force  :  sun's  attraction  on  earth  : :  Mil :  ES ; 

2d.  Sun's  attraction  on  earth :  earth's  attr'n  on  sun : :  354,936  : 1 ; 

3d.  Earth's  attraction  on  sun  :  earth's  attraction  on  moon  :  : 
EM2 :  ES2. 

Compounding  these  three  proportions,  we  have 

Disturbing  force  :  earth's  attraction  on  moon  ::  354,936  x  Mil 
xEM2:ES3. 

Since  the  values  of  Mil,  EM,  and  ES  are  known,  we  can  com- 
pute the  ratio  of  the  disturbing  force  to  the  earth's  attraction. 

Ex.  1.  Compare  the  disturbing  force  of  the  sun  upon  the  moor 


LUNAR  IRREGULARITIES.  147 

with  the  earth's  attraction  upon  the  moon  at  the  time  of  conjunc- 
tion, assuming  the  distance  of  the  sun  to  be  399.32  times  the  dis- 
tance of  the  moon,  and  the  sun's  mass  354,936  times  that  of  the 
earth. 

rig.  T6.  Sun's  att.  on  moon  :  sun's  att.  on 

-  ?  earth  :  :  SE2  :  SM2  :  :  1.00502  :  1. 
Hence,  Disturbing  force  :  sun's  attraction  on  earth  :  :  0.00502  :  1. 
And,  Disturbing  force  :  earth's  attraction  on  moon  ::  354,936  x 

0.00502  :  399.322  :  :  1  :  89  ; 

that  is,  by  the  disturbing  action  of  the  sun  at  conjunction,  the  moon's 
gravity  to  the  earth  is  diminished  by  ^th  part. 

Ex.  2.  Compare  the  disturbing  force  of  the  sun  upon  the  moon 
with  the  earth's  attraction  upon  the  moon  at  the  time  of  opposi- 
tion. 

Fig.  77.  Sun's  att.  on  moon  :  sun's  att.  on 

::SE2:SMa:  :.99501:1. 


Hence,  Disturbing  force  :  sun's  attraction  on  earth  :  :  0.00499  :  1. 
And,    Disturbing  force  :  earth's  attraction  on  moon  ::  354,936  x 

0.00499  :  399.322  :  :  1  :  90  ; 

that  is,  by  the  disturbing  action  of  the  sun  at  opposition,  the  moon's 
gravity  to  the  earth  is  diminished  by  -g^th  part. 

Ex.  3.  Compare  the  disturbing  force  of  the  sun  upon  the  moon 
with  the  earth's  attraction  upon  the  moon  at  the  time  of  quadra- 
ture. 

Disturbing  force  :  sun's  attraction  on  earth  :  :  1  :  399,32.    Art.  269. 
Hence,  Disturbing  force  :  earth's  attraction  on  moon  :  :  354,936  : 

399.323::  1:179; 

that  is,  by  the  disturbing  action  of  the  sun  at  quadrature,  the  moon's 
gravity  to  the  earth  is  increased  by  i^th  part. 

Thus  we  see  that  at  the  quadratures,  the  gravity  of  the  moon 
to  the  earth  is  increased  by  about  the  179th  part,  while  at  the  op- 
position and  conjunction  it  is  diminished  by  about  twice  this  quan- 
tity ;  and,  by  a  computation  extending  to  every  part  of  the  orbit, 
it  is  found  that  the  average  effect  is  to  diminish  the  moon's  gravity  by 
^th  part. 

In  consequence  of  this  diminution  of  her  gravity,  the  moon  de- 
scribes her  orbit  at  a  greater  distance  from  the  earth,  with  a  less 
angular  velocity,  and  in  a  longer  time,  than  if  she  were  urged  to 
the  earth  by  her  gravity  alone. 


ASTRONOMY. 


271.  The  equation  of  the  centre  depends  upon  the  eccentricity  of 
the  orbit     The  eccentricity  of  the  moon's  orbit  was  stated  in 
Art.  208  to  be  -j^th,  and  the  greatest  value  of  the  equation  of  the 
centre  is  6°  18'  17",  being  more  than  three  times  that  of  the  sun. 

272.  Evedion. — After  the  equation  of  the  centre,  the  most  im- 
portant inequality  affecting  the  motion  of  the  moon  is  that  termed 
the  Evection,  the  discovery  of  which  we  owe  to  the  famous  astron- 
omer Hipparchus,  in  the  second  century  before  the  Christian  era. 
The  evection  is  an  inequality  in  the  equation  of  the  centre  de- 
pending on  the  position  of  the  major  axis  of  the  moon's  orbit,  in 
respect  of  the  line  drawn  from  the  earth  to  the  sun. 

273.  Cause  of  evection. — Any  cause  which  at  the  perigee  should 
have  the  effect  to  increase  the  moon's  gravitation  toward  the 
earth  beyond  its  mean,  and  at  the  apogee  to  diminish  the  moon's 
gravitation  toward  the  earth,  would  augment  the  difference  be- 
tween the  gravitation  at  the  perigee  and  apogee,  and,  consequent- 
ly, increase  the  eccentricity  of  the  orbit.    But  any  cause  which  at 
the  perigee  should  have  the  effect  to  diminish  the  moon's  gravi- 
tation toward  the  earth  beyond  its  mean,  and  at  the  apogee  to  in- 
crease it,  would  diminish  the  difference  between  the  two,  and, 
consequently,  diminish  the  eccentricity. 

rig.  TS.        SB  Let  E  represent  the  earth T 

ABCD  the  moon's  orbit,  of 
which  A  is  the  perigee  and 
C  the  apogee,  and  let  SS' 
S"S'"  be  the  apparent  orbit 
of  the  sun.  If  the  sun  be  at 
S,  so  that  the  major  axis  of 
the  moon's  orbit  is  directed 
to  the  sun,  the  distance  of 
the  moon  at  A  from  the 
earth  is  less  than  if  it  moved 
in  a  circle,  and  the  sun's  dis- 
turbing force,  computed  as 
in  Ex.  1,  Art  270,  will  be 
found  to  be  less  than  -g^th  of  the  moon's  gravity.  So,  also,  the 
distance  of  the  moon  from  the  earth  at  C  is  greater  than  if  it 
moved  in  a  circle,  and  the  disturbing  force  computed,  as  in  Ex.  2, 


LUNAR  IRREGULARITIES.  149 


Art.  270,  will  be  found  to  be  greater  than  ^th  part  of  the  moon's 
gravity  ;  that  is,  when  the  transverse  axis  of  the  moon's  orbit  is 
directed  to  the  sun,  the  moon's  gravity  to  the  earth  when  at  peri- 
gee is  diminished  less  than  the  mean,  and  at  apogee  is  diminished 
more  than  the  mean.  Hence  the  moon,  when  at  perigee,  is  drawn 
away  from  the  earth  by  less  than  the  mean  quantity,  and  when 
at  apogee,  is  drawn  away  from  the  earth  by  more  than  the  mean 
quantity.  Thus  the  inequality  between  the  two  distances  of  the 
moon  from  the  earth  is  increased;  that  is,  the  eccentricity  of  the 
moon's  orbit  is  increased. 

But  if  the  sun  be  at  S'  and  the  moon  at  A,  the  sun's  disturbing 
force,  computed  as  in  Ex.  3,  Art.  270,  will  be  found  to  be  less  than 
•p^th  part  of  the  moon's  gravity  ;  but  if  the  moon  be  at  C,  and 
the  sun  at  S',  the  disturbing  force  of  the  sun  will  be  found  to  be 
greater  than  -rf^th  part  of  the  moon's  gravity  ;  that  is,  when  the 
line  of  the  apsides  is  in  quadrature,  the  gravitation  at  the  apogee 
is  most  augmented,  and  that  at  perigee  is  least  augmented.  Hence 
the  effect  of  the  sun's  action  is  to  diminish  the  inequality  between 
the  two  distances  of  the  moon  from  the  earth  at  these  two  points  ; 
that  is,  to  diminish  the  eccentricity  of  the  orbit.  Thus  we  find,  in 
general,  that  the  moon's  orbit  is  most  eccentric  when  the  line  of  the 
apsides  is  in  syzygy,  and  is  least  eccentric  when  the  line  of  the  apsides 
is  in  quadrature.  The  greatest  value  of  evection  is  1°  16'  27//. 

274.  Variation.  —  Another  large  inequality  in  the  moon's  mo- 
tions is  called  the  Variation.     By  comparing  the  moon's  observed 
place  with  the  place  computed  from  the  mean  motion,  the  equa- 
tion of  the  centre,  and  the  evection,  Tycho  Brahe,in  the  sixteenth 
century,  discovered  that  the  two  places  did  not  generally  agree. 
They  agreed  only  at  the  syzygies  and  quadratures,  and  varied 
most  in  the  octants,  where  the  inequality  amounted  to  39'  30". 

275.  Cause  of  variation.  —  This  inequality  is  occasioned  by  that 
part  of  the  sun's  disturbing  force  which  acts  in  the  direction  of  a 
tangent  to  the  moon's  orbit,  Art.  269.    This  force  is  nothing  at  the 
syzygies  and  quadratures,  and  is  greatest  near  the  octants.     It  ac- 
celerates the  moon's  motion  in  going  from  quadrature  to  conjunc- 
tion ;  and  when  the  moon  is  past  conjunction,  the  tangential  force 
changes  its  direction  and  retards  the  moon's  motion. 


150  ASTRONOMY. 

276.  The  annual  equation  is  an  inequality  in  the  moon's  motion 
arising  from  the  variation  of  the  sun's  distance  from  the  earth. 
When  the  earth  is  at  perihelion,  the  sun's  disturbing  force  is 
greater  than  its  average  value ;  the  moon's  gravity  to  the  earth 
is  diminished  more  than  usual ;  and  its  velocity  is  therefore  slow- 
er than  the  mean.     For  the  same  reason,  at  aphelion  the  moon's 
velocity  is  greater  than  the  mean.     The  period  of  this  inequality 
is  one  year,  and  its  maximum  effect  upon  the  moon's  longitude 
amounts  to  11'  9". 

277.  Other  inequalities  in  Hie  moon's  motion. — These  three  ine- 
qualities, evection,  variation,  and  annual  equation,  are  the  largest 
of  the  inequalities  in  the  moon's  motion.     The  other  inequalities 
are  more  minute;  but,  in  order  to  represent  the  moon's  place  with 
the  greatest  possible  accuracy,  it  is  necessary  to  take  into  account 
a  large  number  of  corrections. 

The  moon's  place  for  every  hour  of  the  year  is  computed  sev- 
eral years  beforehand,  and  published  in  the  Nautical  Almanac. 
These  places  are  now  computed  from  Tables  published  by  Pro- 
fessor Hansen  in  1858.  The  average  difference  between  the  ob- 
served places  of  the  moon  and  the  places  computed  from  these 
Tables  does  not  exceed  6",  and  only  once  or  twice  in  a  year  does 
the  difference  amount  to  so  large  a  quantity  as  12". 

278.  Cause  of  the  retrograde  motion  of  the  moon's  nodes.  —  The 
plane  of  the  moon's  orbit  is  inclined  to  the  ecliptic  about  5°;  that 
is,  in  half  of  her  revolution  she  is  on  the  north  side  of  the  eclip- 
tic, and  in  half  is  on  the  south  side  of  the  ecliptic.     The  sun  is 
seldom  in  the  plane  of  the  moon's  orbit,  and  his  action  generally 
has  a  tendescy  to  draw  the  rnoon  out  of  the  plane  in  which  she 
is  moving.     This  oblique  force  may  be  resolved  into  two  other 
forces— one  lying  in  the  plane  of  the  ecliptic,  and  the  other  per- 
pendicular to  it.    Let  ENN"  represent  the  ecliptic,  and  AN  a 

Fig.  79.  A   portion  of  the  moon's 

orbit.  Let  the  moon 
be  at  A,  and  approach- 
ing the  descending 
node  N.  The  sun  be- 
ing situated  in  the 
plane  EN",  his  attrac- 


LUNAR   IRREGULARITIES.  151 

tion  tends  to  draw  the  moon  toward  that  plane.  Let  that  part 
of  the  sun's  disturbing  force  which  is  perpendicular  to  the  plane 
EN  be  represented  by  AB,  and  suppose  that  in  the  time  that  the 
perpendicular  force  would  cause  it  to  describe  AB,  the  moon,  if 
undisturbed,  would  have  advanced  from  A  to  D.  By  the  com- 
bined action  of  these  two  forces,  the  moon  will  describe  the  diag- 
onal AC,  and  cross  the  ecliptic  in  the  point  N'.  Thus  the  node 
has  shifted  from  N  to  N'  in  a  direction  contrary  to  that  of  the 
moon's  motion,  and  the  inclination  of  the  orbit  to  the  ecliptic  has 
increased.  After  the  moon  has  crossed  the  ecliptic,  the  sun's  dis- 
turbing action  tends  to  draw  the  moon  northward  toward  the 
ecliptic.  Suppose  the  moon  to  be  at  F,  and  let  that  part  of  the 
sun's  disturbing  force  which  is  perpendicular  to  the  ecliptic  be 
represented  by  FK,  while  FG  represents  the  moon's  velocity  in 
her  orbit.  The  resultant  of  these  two  forces  will  be  a  motion  in 
the  diagonal  FII,  as  if  the  moon  had  come,  not  from  N',  but  from 
N",  a  point  still  farther  to  the  westward.  Thus  the  node  has 
traveled  farther  westward,  but  the  inclination  of  the  orbit  to  the 
plane  of  the  ecliptic  has  diminished.  Thus  it  appears  that  both 
in  approaching  the  node,  and  in  receding  from  it,  the  node  shifts  its 
place  in  a  direction  contrary  to  that  of  the  moon's  motion;  but  the 
inclination  of  the  moon's  orbit  increases  while  the  moon  approaches 
the  node,  and  diminishes  while  the  moon  is  receding  from  it. 

When  the  line  of  the  nodes  of  the  moon's  orbit  passes  through 
the  sun,  there  is  no  disturbing  force  tending  to  draw  the  moon 
out  of  the  plane  of  its  orbit ;  but  in  every  other  position  the  line 
of  the  nodes  is  constantly  regressing,  making  a  complete  revolu- 
tion in  about  19  years.  See  Art.  240.  The  inclination  of  the 
plane  of  the  orbit  to  the  ecliptic  increases  and  diminishes  al- 
ternately. This  variation  is,  however,  confined  within  very  nar- 
row limits,  so  that  there  is  no  permanent  change  in  the  inclina- 
tion of  the  orbit. 

279.  Cause  of  the  progression  of  the  line  of  the  apsides. — The  ap- 
sides of  the  moon's  orbit  are  distant  from  each  other  more  than 
180°.  This  is  caused  by  the  disturbing  action  of  the  sun,  which 
tends  to  diminish  the  moon's  gravity  to  the  earth.  If  the  moon 
was  only  acted  upon  by  the -earth's  attraction,  she  would  describe 
an  ellipse,  and  her  angular  motion  from  perigee  to  apogee  would 
be  just  180° ;  but  when  the  effect  of  the  sun's  action  is  to  dimin- 


152  ASTRONOMY. 

ish  the  moon's  gravity,  she  will  continually  recede  from  the  ellipse 
that  would  otherwise  be  described;  her  path  will  be  less  curved, 
and  she  must  move  through  a  greater  distance  before  the  radius 
vector  intersects  the  path  at  right  angles.  She  must  therefore 
move  through  a  greater  angular  distance  than  180°  in  going  from 
perigee  to  apogee,  and,  consequently,  the  apsides  must  advance. 
On  the  contrary,  when  the  moon's  gravity  is  increased  by  the 
sun's  action,  her  path  will  fall  within  the  ellipse  which  she  would 
otherwise  describe ;  its  curvature  will  be  increased,  and  the  dis- 
tance through  which  she  must  move  before  the  radius  vector  in- 
tersects her  path  at  right  angles  will  be  less  than  180°.  The  ap- 
sides will  therefore  move  backward.  Now  it  has  been  shown 
that  the  sun's  action  alternately  increases  and  diminishes  the 
moon's  gravity  to  the  earth.  The  motion  of  the  apsides  will 
therefore  be  alternately  direct  and  retrograde.  But  as  the  dimi- 
nution of  gravity  has  place  during  a  much  longer  part  of  the 
moon's  revolution,  and  is  also  greater  than  the  increase,  the  di- 
rect motion  will  exceed  the  retrograde ;  and  in  one  revolution  of 
the  moon,  the  apsides  have  a  progressive  motion  of  about  3°,  mak- 
ing a  complete  revolution  in  about  nine  years.  See  Art.  239. 

280.  Periodical  and  secular  inequalities. — The  perturbations  in 
the  elliptic  movements  of  the  planets  and  their  satellites  may  be 
divided  into  two  distinct  classes.     Those  of  the  first  class  depend 
simply  on  the  configurations  of  the  planets,  and  complete  the  cycle 
of  their  values  upon  each  successive  return  of  the  same  configu- 
ration.   These  are  called  periodic  inequalities.    Their  periods,  gen- 
erally speaking,  are  not  long ;  and  their  general  effect  is  slightly 
to  accelerate  or  retard  a  planet  in  its  orbit.     The  perturbations 
of  the  second  class  depend  on  the  configuration  of  the  nodes  and 
perihelia.     They  vary  with  extreme  slowness,  requiring  centu- 
ries to  complete  the  cycle  of  their  values,  and  they  are  hence  de- 
nominated secular  inequalities.    Laplace  has  indeed  demonstrated 
that  the  last-mentioned  inequalities  are  also  periodic,  but  the  pe- 
riods are  much  longer  than  those  of  the  other  inequalities,  and 
are  independent  of  the  mutual  configurations  of  the  planets. 

281.  Secular  acceleration  of  the  moon's  mean  motion. — The  mean 
motion  of  the  moon  exhibits  a  secular  inequality  which  has  be- 
come very  celebrated.     By  comparing  the  results  of  recent  ob- 


ECLIPSES  OF  THE   MOON.  153 

servations  with  the  Chaldean  observations  of  eclipses  at  Babylon 
in  the  years  719  and  720  before  the  Christian  era,  Dr.  Halley  dis- 
covered that  the  periodic  time  of  the  moon  is  now  sensibly  short- 
er than  at  the  time  of  the  Chaldean  eclipses.  The  mean  motion 
of  the  moon  increases  at  the  rate  of  more  than  10"  in  one  hund- 
red years.  If  this  acceleration  of  her  motion,  and  the  consequent 
diminution  of  her  distance,  were  perpetually  to  continue,  it  would 
follow  that  she  would  eventually  be  precipitated  to  the  earth. 
But  Laplace  has  shown  that  this  acceleration  of  the  moon  is  occa- 
sioned by  the  change  in  the  eccentricity  of  the  earth's  orbit.  It 
has  been  stated,  Art.  113,  that  the  eccentricity  of  the  earth's  orbit 
has  been  diminishing  from  the  time  of  the  earliest  observations. 

The  mean  action  of  the  sun  upon  the  moon  tends  to  diminish 
the  moon's  gravity  to  the  earth,  and  thereby  causes  a  diminution 
of  her  angular  velocity.  This  diminution  being  once  supposed 
to  occur,  the  angular  velocity  would  afterward  remain  constant, 
provided  the  mean  solar  action  always  retained  the  same  value. 
This,  however,  is  not  the  case,  for  it  depends,  to  a  certain  extent, 
on  the  eccentricity  of  the  earth's  orbit.  Now  the  eccentricity  of 
the  earth's  orbit  has  been  continually  diminishing  from  the  date 
of  the  earliest  recorded  observations  down  to  the  present  time ; 
hence  the  sun's  mean  action  must  also  have  been  diminishing, 
and,  consequently,  the  moon's  mean  motion  must  have  been  con- 
tinually increasing.  This  acceleration  will  continue  as  long  as 
the  earth's  orbit  is  approaching  toward  a  circular  form ;  but  as 
soon  as  the  eccentricity  begins  to  increase,  the  sun's  mean  action 
will  increase,  and  the  acceleration  of  the  moon's  mean  motion  will 
be  converted  into  a  retardation. 


CHAPTER  X. 

ECLIPSES  OF  THE   MOON. 

282.  Cause  of  eclipses. — An  eclipse  of  the  sun  is  caused  by  the 
moon  passing  between  the  sun  and  the  earth.  It  can  therefore 
only  occur  when  the  moon  is  in  conjunction  with  the  sun ;  that 
is,  at  the  time  of  new  moon.  An  eclipse  of  the  moon  is  caused 
by  the  earth  passing  between  the  sun  and  moon.  It  can  there- 
fore only  occur  when  the  moon  is  in  opposition ;  that  is,  at  the 
time  of  full  moon. 


154  ASTRONOMY. 

283.  Why  ecli})ses  do  not  occur  every  month. — If  the  moon's  orbit 
coincided  with  the  plane  of  the  ecliptic,  there  would  be  an  eclipse 
of  the  sun  at  every  new  moon,  since  the  moon  would  pass  direct- 
ly between  the  sun  and  earth ;  and  there  would  be  an  eclipse  of 
the  moon  at  every  full  moon,  since  the  earth  would  be  directly 
between  the  sun  and  motm.     But  since  the  moon's  orbit  is  in- 
clined to  the  ecliptic  about  5°,  an  eclipse  can  only  occur  when  the 
moon,  at  the  time  of  new  or  full,  is  at  or  near  one  of  its  nodes. 
At  other  times,  the  moon  is  too  far  north  or  south  of  the  ecliptic 
to  cause  an  eclipse  of  the  sun,  or  to  be  itself  eclipsed. 

284.  Form  of  the  earth's  shadow. — Since  the  magnitude  of  the 
sun  is  far  greater  than  that  of  the  earth,  and  both  bodies  are  of  a 
globular  form,  the  earth  must  cast  a  conical  shadow  in  a  direction 
opposite  to  that  of  the  sun.     Let  AB  represent  the  sun,  and  CD 


Fig.  80. 


the  earth,  and  let  the  tangent  lines  AC,  BD  be  drawn,  and  pro- 
duced to  meet  in  F.  Then  CFD  will  represent  a  section  of  the 
earth's  shadow,  and  EF  will  be  its  axis.  If  the  triangle  AFS  be 
supposed  to  revolve  round  the  axis  SF,  the  tangent  CF  will  de- 
scribe the  convex  surface  of  a  cone,  within  the  whole  of  which 
the  light  of  the  sun  must  be  intercepted  by  the  earth. 

285.  The  semi-angle  of  the  cone  of  the  earth's  shadow  is  equal  to  the 
sun's  apparent  semi-diameter,  minus  his  horizontal  parallax. 

In  Fig.  80  the  semi-angle  of  the  cone  of  the  earth's  shadow  is 
'C  or  EFD.  Now  SEB = EFB + EBF ;  that  is,  EFB = SEB  - 
EBD ;  or  half  the  angle  of  the  cone  of  the  earth's  shadow  is  equal 
to  the  sun's  apparent  semi-diameter,  minus  his  horizontal  paral- 
lax. Putting  s  for  the  sun's  semi-diameter,  and  p  for  his  horizon- 
tal parallax,  we  have  the  semi-angle  of  the  earth's  shadow,  EFC 


ECLIPSES  OF  THE  MOON.  155 

286.  The  length  of  the  earth's  shadow  varies  according  to  the  dis- 
tance of  the  sun  from  the  earth;  its  mean  length  being  856,200  miles, 
or  more  than  three  times  the  distance  of  the  moon  from  the  earth. 

In  the  right-angled  triangle  EFC,  right-angled  at  C, 


. 

sin.  Lt  C     sin.  (s—  p) 

The  mean  value  of  the  sun's  apparent  semi-diameter  is  16'  1".8, 
and  the  sun's  horizontal  parallax  is  8".  6  ;  hence  s—  p  =  15'  53'  '.2. 
Also,  the  mean  radius  of  the  earth  =3956.7  miles.  Hence  the 

average  length  of  the  earth's  shadow  =  -.  —  .,,,  '.  ,„  ,=856,200 

sin.  lo  08  .2 

miles. 

Since  the  mean  distance  of  the  moon  from  the  earth  is  only 
238,880  miles,  the  shadow  extends  to  a  distance  more  than  three 
times  that  of  the  moon. 

287.  The  average  breadth  of  the  earth's  shadow,  at  ti^e  distance  of 
Hie  moon,  is  almost  three  times  the  moon's  diameter. 

Let  M'M"  represent  a  portion  of  the  moon's  orbit.  The  appa- 
rent semi-diameter  of  the  earth's  shadow  at  the  distance  of  the 
moon  is  the  angle  MEH.  But  EHD  =  FEH+HFE.  Hence  MEH 
=EHD  —  HFE.  ButEHD  =  the  moon's  horizontal  parallax;  and 
HFE  =  the  sun's  semi-diameter  minus  his  horizontal  parallax 
(=s—  p);  therefore  half  the  angle  subtended  by  the  section  of 
the  shadow  is  equal  to  the  sum  of  the  parallaxes  of  the  sun  and 
moon,  minus  the  sun's  semi-diameter.  If  we  represent  the  moon's 
horizontal  parallax  by  p',  we  shall  have 
MEK—p+p'—s. 

The  mean  value  of  p'  is  57'  2".3,  and  s—p  =  l5r  53".2;  hence 
p+p'—  5=41'  9".l.  The  mean  value  of  the  moon's  apparent 
semi-diameter  is  15'  39".9.  Hence  the  diameter  of  the  shadow  is 
almost  three  times  the  moon's  diameter,  and  therefore  the  moon 
may  be  totally  eclipsed  for  as  long  a  time  as  she  takes  to  describe 
about  twice  her  own  diameter.  The  eclipse  will  begin  when  the 
moon's  disc  at  M'  touches  the  earth's  shadow,  and  the  eclipse  will 
end  when  the  moon's  disc  touches  the  earth's  shadow  at  M". 

288.  Lunar  ecliptic  limits.  —  There  is  a  certain  distance  of  the 
moon's  node  from  the  centre  of  the  earth's  shadow  beyond  which 
a  lunar  eclipse  is  impossible,  and  a  certain  less  distance  within 


156  ASTRONOMY. 

which  an  eclipse  is  inevitable.  These  distances  are  called  the  lunai 
ecliptic  limits.  The  first  is  called  the  major  limit,  and  the  second 
the  minor  limit 

Let  NE   repre" 
sent    the   ecliptic, 

NM  the  moon's 
orbit,  and  N  the 
moon's  ascending 
node.  Let  EA  be 
the  semi-diameter  of  the  earth's  shadow,  and  MA  the  semi-diam- 
eter of  the  moon.  When  the  line  ME,  joining  the  centres  of  the 
moon  and  shadow,  becomes  equal  to  the  sum  of  the  semi-diam- 
eters, the  moon  will  touch  the  earth's  shadow ;  and  if  ME  be  less 
than  this  limit,  the  moon  will  enter  the  shadow,  and  be  partially 
or  totally  eclipsed.  The  line  NE  represents  that  distance  of  the 
moon's  node  from  the  centre  of  the  earth's  shadow  beyond  which 
there  can  be  no  eclipse. 

289.  To  compute  the  values  of  the  ecliptic  limits. — We  may  regard 
EMN  as  a  spherical  triangle,  right-angled  at  M,  in  which  EM  rep- 
resents the  sum  of  the  radii  of  the  moon  and  of  the  earth's  shad- 
ow, and  N  is  the  inclination  of  the  moon's  orbit  to  the  ecliptic- 
Now,  by  Napier's  rule, 

R  sin.  EM=sin.  EN  sin.  N ;  or  sin.  EN=sin'  ™. 

sin.  N 

Since  EM  and  N  are  both  variable,  the  ecliptic  limit  is  varia- 
ble. To  obtain  the  distance  beyond  which  a  lunar  eclipse  is  im- 
possible, we  must  employ  the  greatest  possible  value  of  EM,  and 
the  least  possible  value  of  N.  To  obtain  the  distance  within 
which  an  eclipse  is  inevitable,  we  must  employ  the  least  possible 
value  of  EM,  and  the  greatest  possible  value  of  N.  The  greatest 
possible  value  of  EM  is  62'  38",  and  the  least  inclination  of  the 
moon's  orbit  to  the  ecliptic  is  5°,  from  which  we  obtain  the  major 
limit  of  lunar  eclipses,  12°  4'.  The  least  possible  value  of  EM  is 
52'  20",  and  the  greatest  possible  inclination  of  the  moon's  orbit 
to  the  ecliptic  is  5°  17',  from  which  we  obtain  the  minor  limit  of 
lunar  eclipses,  9°  30'. 

If,  then,  at  the  time  of  opposition,  the  moon's  node  is  distant 
from  the  centre  of  the  earth's  shadow  less  than  9°  30',  or  if  the 
sun  be  distant  from  the  opposite  node  of  the  moon  less  than  9°  30', 


ECLIPSES  OF  THE   MOON.  157 

there  will  certainly  be  an  eclipse  of  the  moon ;  but  if  the  sun  be 
distant  from  the  node  of  the  moon's  orbit  more  than  12°  4',  there 
can  not  be  an  eclipse.  When  the  distance  falls  between  these 
limits,  it  will  be  necessary  to  make  a  more  minute  calculation  in 
order  to  determine  whether  there  will  or  will  not  be  an  eclipse. 

290.  Different  lands  of  lunar  eclipses. — When  the  moon  just 
touches  the  earth's  shadow,  but  passes  by  it  without  entering  it, 
the  circumstance  is  called  an  appulse.     When  a  part,  but  not  the 
whole  of  the  moon  enters  the  shadow,  the  eclipse  is  called  a  par- 
tial eclipse ;  when  the  moon  enters  entirely  into  the  shadow,  it  is 
called  a  total  eclipse;  and  if  the  moon's  centre  should  pass  through 
the  centre  of  the  shadow,  it  would  be  called  a  central  eclipse.     It 
is  probable,  however,  that  a  strictly  central  eclipse  of  the  moon 
has  never  occurred. 

291.  The  earth's  penumbra. — Long  before  the  moon  enters  the 
cone  of  the  earth's  shadow,  the  earth  begins  to  intercept  from  it  a 
portion  of  the  sun's  light,  so  as  to  render  the  illumination  of  its 
surface  sensibly  more  faint.     This  partial  shadow  is  called  the 
earth's  penumbra.     Its  limits  are  determined  by  the  tangent  lines 
AD,  BC  produced.     Throughout  the  space  included  between  the 

Fig.  82. 


lines  CK  and  DL,  the  light  of  the  sun  is  more  or  less  obstructed 
by  the  earth.  If  a  spectator  were  situated  at  L,  he  would  see  the 
entire  disc  of  the  sun ;  but  between  L  and  the  line  DF,  he  would 
see  only  a  portion  of  the  sun's  surface,  and  the  portion  of  the  sun 
which  was  hidden  would  increase  until  he  reached  the  line  DF, 
beyond  which  the  sun  would  be  entirely  hidden  from  view. 

292.  The  semi-angle  of  the  earth's  penumbra  is  equal  to  the  sun's 
apparent  semi-diameter,  plus  his  horizontal  parallax. 
The  an-le  KNF=BNS=BEN  +  NBE.    But  BEN  is  the  sun's 


158  ASTRONOMY. 

apparent  semi-diameter,  and  NBE  is  the  sun's  horizontal  parallax. 
Hence  the  semi-angle  of  the  penumbra  is  represented  by  s+p. 

293.  The  apparent  semi-diameter  of  a  section  of  the  penumbra,  at 
the  moon's  distance  is  equal  to  Hie  sum  of  the  parallaxes  of  the  sun 
and  moon,  plus  the  sun's  semi-diameter. 

The  angle  MEra=ENra+E7«N. 
But  ENm=s+p,  Art.  292. 
And  EmN=the  moon's  horizonal  parallax  =pr. 
Hence  M.Em=p+p'+s,  which  equals  the  apparent  semi-diam- 
eter of  the  shadow,  plus  the  sun's  diameter. 

294.  Effect  of  the  earth's  atmosphere. — In  obtaining  the  above 
expression  for  the  dimensions  of  the  earth's  shadow,  the  shadow 
is  assumed  to  be  limited  by  those  rays  of  the  sun  which  are  tan- 
gents to  the  sun  and  earth.     It  is,  however,  found  that  the  ob- 
served duration  of  an  eclipse  always  exceeds  the  duration  com- 
puted on  this  hypothesis.     This  fact  is  accounted  for  in  part  by 
supposing  that  most  of  those  rays  which  pass  near  the  surface  of 
the  earth  are  absorbed  by  the  lower  strata  of  the  atmosphere;  but 
we  must  also  admit  that  those  rays  of  the  sun  which  enter  the  at- 
mosphere, and  are  so  far  from  the  surface  as  not  to  be  absorbed, 
are  refracted  toward  the  axis  of  the  shadow,  and  are  thus  spread 
over  the  entire  extent  of  the  geometrical  shadow,  thereby  dimin- 
ishing the  darkness,  but  increasing  the  diameter  of  the  shadow, 
and,  consequently,  the  duration  of  the  eclipse. 

In  consequence  of  the  gradual  diminution  of  the  moon's  light 
as  it  enters  the  penumbra,  it  is  difficult  to  determine  with  accu- 
racy the  instant  when  the  moon  enters  the  dark  shadow ;  and  as- 
tronomers have  differed  as  to  the  amount  of  correction  that  should 
be  made  for  the  effect  of  the  earth's  atmosphere.  It  is  generally 
found  necessary,  however,  to  increase  the  computed  diameter  of 
the  shadow  by  about  ^th  part. 

295.  Moon  visible  when  entirely  immersed  in  the  earth's  shadow. — 
When  the  moon  is  totally  immersed  in  the  earth's  shadow,  she 
does  not,  except  on  some  rare  occasions,  become  in  visible,  but  as- 
sumes a  dull  reddish  hue,  somewhat  of  the  color  of  tarnished  cop- 
per.    This  arises  from  the  refraction  of  the  sun's  rays  in  passing 
through  the  earth's  atmosphere,  as  explained  in  the  preceding 


ECLIPSES  OF  THE   MOON. 


159 


Article.  Those  rays  from  the  sun  which  enter  the  atmosphere, 
and  are  so  far  from  the  surface  as  not  to  be  absorbed,  are  bent  to* 
ward  the  axis  of  the  shadow,  and  fall  upon  the  moon,  causing  suf- 
ficient illumination  to  render  the  disc  distinctly  visible. 

296.  Computation  of  lunar  eclipses. — By  the  solar  tables  we  may 
ascertain  the  apparent  position  of  the  centre  of  the  sun  from  hour 
to  hour,  and  hence  we  may  learn  the  position  of  the  centre  of  the 
earth's  shadow.     From  the  lunar  tables  we  ascertain,  in  the  same 
manner,  the  position  of  the  moon's  centre  from  hour  to  hour.    The 
eclipse  will  begin  when  the  distance  between  the  centre  of  the 
moon  and  that  of  the  shadow  is  equal  to  the  sum  of  the  apparent 
semi-diameters  of  the  moon's  disc  and  the  shadow ;  the  middle  of 
the  eclipse  will  occur  when  this  distance  is  least ;  and  the  eclipse 
will  end  when  the  distance  between  the  centres  is  again  equal  to 
the  sum  of  the  apparent  semi-diameters.     The  Nautical  Almanac 
for  each  year  furnishes  the  places  of  the  sun  for  every  day  of  the 
year,  as  computed  from  the  solar  tables,  and  the  places  of  the 
moon  are  given  for  every  hour  of  the  year.     With  this  assist- 
ance, it  is  easy  to  compute  the  times  of  beginning  and  end  of  an 
eclipse. 

297.  Construction  of  the  diagram. — First  find  the  time  of  oppo* 
sition,  or  the  time  of  full  moon.     For  this  time  compute  the  dec* 
lination,  horizontal  parallax,  and  semi-diameter  both  of  the  sun 
and  moon ;  also  the  hourly  motion  of  the  moon  from  the  sun 
both  in  right  ascension  and  declination. 

Let  C  represent  the 
centre  of  the  earth's 
shadow.  Draw  the  line 
ACB  parallel  to  the 
equator,  and  DCM  per- 
pendicular to  it.  Select 
a  convenient  scale  of 
equal  parts,  and  from  it 
take  CG-,  equal  to  the 
moon's  declination,  mi- 
nus the  declination  of 
the  centre  of  the  shad- 
ow, and  set  it  on  CD 


130  ASTRONOMY. 

from  C  to  G,  above  the  line  AB,  if  the  centre  of  the  moon  is  north 
of  the  centre  of  the  shadow,  but  below  if  south.  Take  CO,  equal 
to  the  hourly  motion  of  the  moon  from  the  sun  in  right  ascension, 
reduced  to  the  arc  of  a  great  circle,  and  set  it  on  the  line  CB,  to 
the  right  of  C.  Take  CP,  equal  to  the  moon's  hourly  motion  from 
the  sun  in  declination,  and  set  it  on  the  line  CD  from  C  to  P, 
above  the  line  AB,  if  the  moon  is  moving  northward  with  respect 
to  the  shadow,  but  below  if  moving  southward.  Join  the  points 
O  and  P.  The  line  OP  will  represent  the  hourly  motion  of  the 
moon  from  the  sun ;  and  parallel  to  it,  through  G,  draw  NGL, 
which  will  represent  the  relative  orbit  of  the  moon,  the  earth's 
shadow  being  supposed  stationary.  On  this  line  are  to  be  marked 
the  places  of  the  moon  before  and  after  opposition,  by  means  of 
the  hourly  motion  OP,  in  such  a  manner  that  the  moment  of  op- 
position may  fall  exactly  upon  the  point  G. 

298.  To  determine  the  beginning  and  end  of  the  eclipse. — The  semi- 
diameter  of  the  earth's  shadow  is  equal  to  the  horizontal  parallax 
of  the  moon,  plus  that  of  the  sun,  minus  the  sun's  semi-diameter, 
which  result  must  be  increased  by  -g^th  part,  on  account  of  the 
earth's  atmosphere.  With  this  radius,  describe  the  circle  ADB 
about  the  centre  C.  Add  the  moon's  semi-diameter  to  the  radius 
CB,  and,  with  this  sum  for  a  radius,  describe  about  the  centre  C  a 
circle,  which,  if  there  be  an  eclipse,  will  cut  NL  in  two  points,  E 
and  H,  representing  respectively  the  places  of  the  moon's  centre 
at  the  beginning  and  end  of  the  eclipse.  Draw  the  line  CKR  per- 
pendicular to  LN,  and  cutting  it  in  K.  The  hours  and  minutes 
marked  on  the  line  LN,  at  the  points  E,  K,  and  H,  will  represent 
respectively  the  times  of  the  beginning  of  the  eclipse,  middle  of 
the  eclipse,  and  end  of  the  eclipse.  If  the  circle  does  not  inter- 
sect NL,  there  will  be  no  eclipse.  With  a  radius  equal  to  the 
moon's  semi-diameter,  describe  a  circle  about  each  of  the  centres 
E,  H,  and  K.  If  the  eclipse  is  total,  the  whole  of  the  circle  about 
K  will  fall  within  ARB ;  but  if  part  of  the  circle  falls  without 
ARB,  the  eclipse  will  be  partial.  In  either  case,  the  magnitude 
of  the  eclipse  will  be  represented  by  the  ratio  of  the  obscured  part 
HI  to  the  moon's  diameter.  When  the  eclipse  is  total,  the  begin- 
ning and  end  of  total  darkness  may  be  found  by  taking  a  radius 
equal  to  CB,  diminished  by  the  moon's  semi-diameter,  and  describ- 
ing with  it  round  the  centre  C  a  circle  cutting  LN  in  two  points, 


ECLIPSES  OF  THE   MOON. 


161 


representing  respectively  the  places  of  the  moon's  centre  at  the 
beginning  and  end  of  total  darkness. 

Example  1. ' 

299.  Required  the  times  of  beginning,  end,  etc.,  of  the  eclipse 
of  the  moon,  March  30, 1866,  at  Greenwich. 

By  the  Nautical  Almanac,  the  Greenwich  mean  time  of  oppo- 
sition in  right  ascension  is,  March  30, 16h.  39m.  18.9s.  Corre- 
sponding to  this  time,  the  Nautical  Almanac  furnishes  the  fol- 
lowing elements : 

Declination  of  the  moon S.  4C 


Declination  of  the  earth's  shadow     -     - 
Moon's  equatorial  horizontal  parallax   - 

Sun's  horizontal  parallax 

Moon's  semi-diameter 

Sun's  semi-diameter 

Moon's  hourly  motion  in  right  ascension 
Sun's  hourly  motion  in  right  ascension 
Hourly  motion  of  moon  in  declination 
Hourly  motion  of  shadow  in  declination 


S.4 


12'  55".5 
3   42  .3 
54   28  .1 


14 

16 

28 
2 


52  .0 
2  .2 
48  .0 
16  .4 
14  .1 
58  .1 


The  figure  of  the  earth  being  spheroidal,  that  of  the  shadow 
will  deviate  a  little  from  a  circle,  so  that  instead  of  the  equatorial 
horizontal  parallax,  we  should  employ  the  horizontal  parallax  be- 
longing to  the  mean  latitude  of  45°.  The  reduction  for  latitude, 

Fig.  84. 


162  ASTRONOMY. 

by  Table  VIII. ,  is  5".4,  so  that  the  moon's  reduced  parallax  is 
54'  22".7.  Then,  to  obtain  CB,  the  semi-diameter  of  the  earth's 
shadow,  we  have  54'  22".7+8".6-16'  2".2,  which  is  equal  to  38' 
29".l.  Increasing  this  by  ^ib.  part  of  itself,  or  38".5,  we  have 
39'  7".6  =  CB ;  to  which  adding  the  moon's  semi-diameter,  we  ob- 
tain CE=53'  59".6.  From  the  centre  C,  with  a  radius  CB,  taken 
from  a  convenient  scale  of  equal  parts,  describe  the  circle  ARB, 
representing  the  earth's  shadow.  Draw  the  line  ACB  to  repre- 
sent a  parallel  to  the  equator,  and  make  CG  perpendicular  to  it, 
equal  to  9'  13".2,  which  is  the  moon's  declination,  minus  the  dec- 
lination of  the  centre  of  the  shadow ;  the  point  G  being  taken 
below  C,  because  the  centre  of  the  moon  is  south  of  the  centre 
of  the  shadow. 

The  hourly  motion  of  the  moon  from  the  sun  in  right  ascension 
is  26'  31".6,  which  must  be  reduced  to  the  arc  of  a  great  circle  by 
multiplying  it  by  the  cosine  of  the  moon's  declination,  4°  12'  55", 
Art.  152,  thus :  26'  31".6  =  1591".6  =  3.201834 

cos.dec.4°12'55'/    =9.998824 
Reduced  hourly  motion  =  1587".3  =  3.200658 

Make  CO  =  1587".3,  and  CP,  perpendicular  to  it,  equal  to  8' 
16".0,  which  is  the  hourly  motion  of  the  moon  from  the  shadow 
in  declination,  the  point  P  being  placed  below  C,  because  the4 
moon  was  moving  southward  with  respect  to  the  shadow.  Join 
OP ;  and  parallel  to  it,  through  G,  draw  the  line  NGL,  which  rep- 
resents the  path  of  the  moon  with  respect  to  the  shadow.  On 
NL  let  fall  the  perpendicular  CK.  Now  at  16h.  39m.  18.9s.  the 
moon's  centre  was  at  G.  To  find  X,  the  place  of  the  moon's  cen- 
tre at  16h.,  we  must  institute  the  proportion 

60m. :  39m.  18.9s.  ::OP:GX; 

which  distance,  set  on  the  line  GN",  to  the  right  of  G,  reaches  to 
the  point  X,  where  the  hour,  16h.  preceding  the  full  moon,  is  to 
be  marked.  Take  the  line  OP,  and  lay  it  from  16h.,  toward  the 
right  hand,  to  15h.,  and  successively  toward  the  left  to  17h.,  18h., 
etc.  Subdivide  these  lines  into  60  equal  parts,  representing  min- 
utes, if  the  scale  will  permit;  and  the  times  corresponding  to  the 
points  E,  K,  and  H  will  represent  respectively  the  beginning  of 
the  eclipse,  14h.  38m. ;  the  middle  of  the  eclipse,  16h.  33m. ;  and 
the  end  of  the  eclipse,  18h.  28m. 

If  the  results  obtained  by  this  method  are  not  thought  to  be 
sufficiently  accurate,  we  may  institute  a  rigorous  computation. 


ECLIPSES  OF  THE  MOON.  163 


Computation  of  the  Eclipse. 

300.  The  phases  of  the  eclipse  may  be  accurately  computed  in 
the  following  manner : 

In  the  right-angled  triangle  OCP,  we  have  given  C0  =  1587".3, 
and  CP=496".0,  to  find  OP  and  the  angle  CPO,  thus: 
CP :  K : :  CO  :  tang.  CPO. 

CO =1587.3  =  3.200658 
CP=  496.0=2.695482 
CPO  =  72°  38'  49"  tang.  =  0.505176 
Also,  sin.  CPO  :K::  CO:  OP. 

C0=3.200658 

sin.CPO  =  9.979769 

OP=1663".0  =  3.220889 

301.  Beginning,  middle,  and  end  of  the  eclipse. — The  middle  of 
the  eclipse  is  found  by  means  of  the  triangle  CGK,  which  is  simi- 
lar to  CPO,  because  EG  and  OP  are  parallel,  and  CK  is  perpen- 
dicular to  PO.    Hence  the  angle  CGK  =  72°  38'  49" ;  and  CG,  the 
difference  of  declination  between  the  moon  and  the  centre  of  the 
shadow  =  9'  13'/.2=553//.2.     To  find  CK  and  KG,  we  have  the 
proportions 

E :  CG : :  sin.  CGK  :  CK : :  cos.  CGK :  GK. 
sin.  CGK =9.979769  cos.  CGK =9.474593 

CG=2.742882  CG=2.742882 

CK=528".0=2.722651  GK=165".0= 2.217475 

Then,  to  find  the  time  of  describing  GK,  we  say, 
As  OP  (1663".0)  is  to  GK  (165".0),  so  is  1  hour  to  the  time 
(357.2s.),  5m.  57.2s.,  between  the  middle  of  the  eclipse  and  the 
time  of  opposition  in  right  ascension,  16h.  39m.  18.9s.,  which  gives 
the  time  of  middle  of  the  eclipse  16h.  33m.  21.7s. 

Now,  in  the  triangle  CKE,  we  have  the  hypothenuse  CE=53' 
59".6=3239".6,  and  CK=528".0,  to  find  KE,  thus: 

KE=N/CE2-CK2=v/CE+C^xCE-CK=3196".3. 

To  find  the  time  of  describing  KE,  we  form  the  proportion 

1663".0  :  3196".3  ::  3600s. :  6919.3s.  =  lh.  55m.  19.3s. ; 
which,  subtracted  from  16h.  33m.  21.7s.,  the  time  of  middle,  gives 
14h.  38m.  2.4s.  for  the  beginning  of  the  eclipse ;  and,  added  to  the 
time  of  middle,  gives  for  the  end  of  the  eclipse  18h.  28m.  41.0s. 


164  ASTRONOMY. 

302.  Magnitude  of  the  eclipse.— Subtracting  CK,  8'  48".0,  from 
CR,  39'  7".6,  we  have  KR,  30'  19".6 ;  to  which  adding  KI,  14' 
52".0,  we  obtain  El,  45'  11".6.     Dividing  this  by  the  moon's  di- 
ameter, 29'  44".0,  we  obtain  the  magnitude  of  the  eclipse,  1.520 
(the  moon's  diameter  being  unity) ;  and  the  eclipse  takes  place  on 
the  moon's  north  limb. 

The  magnitude  of  an  eclipse  is  sometimes  expressed  in  digits, 
or  twelfths  of  the  moon's  diameter.  In  the  present  instance,  the 
eclipse  amounts  to  18  digits. 

303.  Beginning  and  end  of  total  darkness. — The  beginning  and 
end  of  total  darkness  may  be  found  in  the  same  manner.     With 
a  radius  equal  to  CB,  diminished  by  the  moon's  semi-diameter 
(that  is,  39'  7".6— 14'  52".0,  which  equals  24'  15".6,  or  1455'/.6), 
describe  about  the  centre  C  a  circle  cutting  LN  in  the  points  S 
and  T,  which  will  represent  the  places  of  the  moon's  centre  at  the 
beginning  and  end  of  total  darkness. 

In  the  triangle  CKS,  CS=1455".6,  and  CK=528".0. 

Hence  KS=v/1455.62— 528.02=1356".5. 

Then,  to  find  the  time  of  describing  KS,  we  say, 

1663".0  :  1356".5  : :  3600s. :  2936.4s. =48m.  56.4s. ; 
which,  being  subtracted  from  16h.  33m.  21.7s.,  gives  the  beginning 
of  total  darkness  15h.  44m.  25.3s. ;  and,  being  added  to  the  time 
of  middle,  gives  for  the  end  of  total  darkness  17h.  22m.  18.1s. 

304.  Contacts  with  Hie  penumbra. — The  contacts  with  the  penum- 
bra may  be  found  in  a  similar  manner.     The  semi-diameter  of  the 
penumbra  is  equal  to  the  semi-diameter  of  the  shadow,  plus  the 
sun's  diameter,  or  39'  7".6  +  32'  4".4  =  71'  12".0.     If  we  take  the 
circle  ARB,  Fig.  84,  to  represent  the  limits  of  the  penumbra,  CE 
will  be  equal  to  71'  12".0  +  14'  52".0=86'  4".0. 

Then,  in  the  triangle  CKE,  we  have  given  CE=5164".0,  and 
CK=528".0. 

Hence  KE=v/51t>4V!-5282=5136".9. 

To  find  the  time  of  describing  KE,  we  sav, 

1663".0 :  5136".9  : :  3600s. :  11120.3s.  =3h.  5m.  20.3s. ; 
which,  being  subtracted  from  16h.  33m.  21.7s.,  gives  the  first  con- 
tact with  the  penumbra  at  13h.  28rn.  1.4s. ;  and,  being  added  to 
the  time  of  middle,  gives  for  the  last  contact  with  the  penumbra 
19h.  38m.  42.0s. 


ECLIPSES   OF  THE   MOON.  165 

305.  Results. — The  results  thus  obtained  are  as  follows : 
First  contact  with  the  penumbra  at  13h.  28rn.  1.4s.  "1 
First  contact  with  the  umbra     -     -  1-4    38       2.5 
Beginning  of  total  eclipse     -    -    -  15    44    25.3       Meantime 

Middle  of  the  eclipse 16    33    21.7     •        at 

End  of  total  eclipse 17    22    18.1       Greenwich. 

Last  contact  with  the  umbra      -     -  18     28     40.9 

Last  contact  with  the  penumbra     -  19     38     42.0 
Magnitude  of  the  eclipse,  1.520  on  the  northern  limb. 

306.  Times  for  any  other  meridian. — To  obtain  the  times  for  any 
other  place,  we  have  only  to  add  or  subtract  the  longitude.     For 
New  Haven,  whose  longitude  is  4h.  51m.  41.6s.  west  of  Greenwich, 
the  times  will  accordingly  be 

First  contact  with  the  penumbra  at     8h.  36m.  20s. 

First  contact  with  the  umbra     -     -     9     46     21 

Beginning  of  total  eclipse      -     -     -  10    52      44         Mean  time 

Middle  of  the  eclipse 11     41      40  at 

End  of  total  eclipse 12     30      37       New  Haven. 

Last  contact  with  the  umbra      -     -  13     36      59 
Last  contact  with  the  penumbra     -  14    47       0 

Ex.  2.  Compute  the  phases  of  the  eclipse  of  June  11, 1881, 
for  New  York  city,  longitude  4h.  56m.  0.2s.  west  of  Greenwich, 
from  the  following  elements : 

Greenwich  mean  time  of  opposition  in  right 

ascension 18h.  54m.  23.3s. 

Declination  of  the  moon S.  22°  52'  52".S 

Declination  of  the  earth's  shadow  -    -    -  S.  23     10  26  .2 
Moon's  equatorial  horizdntal  parallax   -    -  60  33  .4 

Sun's  horizontal  parallax 8  .8 

Moon's  semi-diameter 1631. 7 

Sun's  semi-diameter  -    -     - 15   46  .9 

Moon's  hourly  motion  in  right  ascension   -  40    20  .6 

Sun's  hourly  motion  in  right  ascension-    -  2   35  .6 

Hourly  motion  of  moon  in  declination  -    -          N.  1      0  .9 

Hourly  motion  of  shadow  in  declination    -  S.  0      9  .3 

Ans.  First  contact  with  the  penumbra  at  llh.  18.8m. "] 

First  contact  with  the  umbra  -    -    -    -  12     14.7      I  Mean  time 

Middle  of  the  eclipse 13    57.4      L        at 

Last  contact  with  the  umbra  -     -    -     -  15     40.1      |  New  York. 
Last  contact  with  the  penumbra  -    -    -  16     36.0     J 


166  ASTRONOMY. 


CHAPTER  XL 

ECLIPSES  OF  THE  SUN. 

307.  Length  of  the  moon's  shadow. — The  length  of  the  moon's 
shadow  is  about  equal  to  the  distance  of  the  moon  from  the  earth, 
being  alternately  a  little  greater  and  a  little  less. 

Suppose  the  moon  at  conjunction  to  be  at  one  of  her  nodes. 
Her  centre  will  then  be  in  the  plane  of  the  ecliptic,  and  in  the 
straight  line  passing  through  the  centres  of  the  sun  and  earth. 

^  Fig.  85. 


Let  ASB  be  a  section  of  the  sun,  KEL  that  of  the  earth,  and 
CMD  that  of  the  moon  interposed  directly  between  them.  Draw 
AC,  BD,  tangents  to  the  sun  and  moon,  and  produce  these  liu.es 
to  meet  in  V.  Then  Y  is  the  vertex  of  the  moon's  shadow  ;  and 
these  lines  represent  the  outlines  of  a  cone,  whose  base  is  AB,  and 
whose  vertex  is  V. 

The  angle  8MB  =  MVB  +  MBV  ; 

hence  MVB  =  8MB  -  MBV. 

But      8MB  :  SEB  :  :  SE  :  SM  (Art.  1  1  1)  :  :  400  :  399  ; 

therefore  * 


Now  SEB,  the  sun's  mean  semi-diameter  as  seen  from  the  earth 
=  16'  1".8;  hence  8MB  =  16'  4".2,  which  is  the  sun's  semi-diam- 
eter as  seen  from  the  moon. 

Put  j>=the  sun's  horizontal  parallax. 

_p'=the  moon's  horizontal  parallax. 
s'=the  moon's  semi-diameter. 

Since  the  parallaxes  of  bodies  at  different  distances  are  inverse- 
ly as  the  distances,  Art.  136,  we  shall  have 


ECLIPSES  OF  THE  SUN.  167 

But  since  the  apparent  diameters  of  the  same  body  at  different 
distances  are  inversely  as  the  distances,  Art  111,  we  shall  have 
SM:ME::*':MBC; 

hence  MBO=-^-. 

y-jp 

Now  p,  p',  and  s'  are  known  quantities;  hence  MBC=2".3, 
which  is  the  sun's  horizontal  parallax  as  observed  from  the  moon. 

Hence  MVB,  the  semi-angle  of  the  cone  of  the  moon's  shadow, 
equals  16'  4".2-2".3  =  16'  1".9. 

Then  sin.  16'  1".9  :  DM  (1080  miles)  ::  rad. :  MV  =  231,590 
miles. 

But  the  mean  distance  of  the  moon  from  the  earth's  centre  is 
238,883  miles.  Hence,  when  the  moon  is  at  the  mean  distance 
from  the  earth,  her  shadow  will  not  quite  reach  to  the  earth's  sur- 
face. 

When  the  earth  is  at  its  greatest  distance  from  the  sun,  the 
sun's  apparent  semi-diameter  is  15'  45  ".5 ;  and  the  angle  MVB= 
15'  45".  6. 

-In  this  case  MV= 235,582  miles.  Now  when  the  moon  is  near- 
est the  earth,  her  distance  from  the  centre  of  the  earth  is  only 
221,436  miles.  Hence,  when  the  moon  is  nearest  to  us,  and  her 
shadow  is  the  longest,  the  shadow  extends  14,000  miles  beyond 
the  earth's  centre,  or  about  three  and  a  half  times  the  earth's  ra- 
dius ;  and  there  must  be  a  total  eclipse  of  the  sun  at  all  places 
within  this  shadow. 

308.  Breadth  of  the  moon's  shadow  at  the  earth. — The  greatest 
breadth  of  the  moon's  shadow  at  the  earth,  when  it  falls  perpen- 
dicularly on  the  surface,  is  about  166  miles. 

In  the  triangle  FEV,      FE  :  EV : :  sin.  FVE  :  sin.  VFE. 

But  when  the  moon  is  nearest,  and  the  shadow  is  the  longest, 
EV=14,146  miles;  and  the  angle  FVE  =  15'  45'/.6.  Also,  FE 
=3956.6  miles. 

In  this  case  VFE =56' 20".9. 

But  the  angle  FEG  =  VFE+FVE  =  56'  20".9  +  15'  45".6  = 
1°  12'  6".5  =  arc  FG.  Hence  the  arc  FH=2°  24'  13";  and  if 
we  allow  69  miles  to  a  degree,  the  breadth  of  the  moon's  shadow 
is  166  miles,  nearly. 

When  the  moon  is  at  some  distance  from  the  node,  the  shadow 
falls  obliquely  on  the  earth,  and  its  greatest  breadth  will  evident- 
ly be  increased. 


168  ASTRONOMY. 

309.  Breadth  of  tJie  moon's  penumbra  at  the  earth. — The  greatest 
breadth  of  the  moon's  penumbra  at  the  earth's  surface,  when  it 
falls  perpendicularly  on  the  surface,  is  about  4800  miles. 

If  we  draw  the  tangent  lines  AD,  BC,  and  produce  them  to 
meet  the  earth,  the  sun's  rays  will  be  partially  excluded  from 
the  space  included  between  DV  and  DL,  as  also  between  CV  and 
CK.  Any  point  on  the  line  CK  will  receive  light  from  all  points 
of  the  sun's  disc.  As  the  point  advances  toward  CV,  it  will  re- 
ceive less  and  less  of  the  sun's  light,  since  a  larger  portion  of  the 
moon,  M,  will  be  interposed  between  it  and  the  sun.  At  the 
boundary  CV,  all  the  rays  of  the  sun  are  intercepted.  This  space, 
KCV,  from  which  the  sun's  light  is  partially  intercepted,  is  called 
the  moon's  pen  umbra. 

The  semi-angle  of  the  penumbra  CIM  =  SCB  +  CSM,  of  which 
SCB  is  the  sun's  apparent  semi-diameter  at  the  moon,  and  CSM 
is  the  sun's  horizontal  parallax  at  the  moon.  The  breadth  of  the 
penumbra  will  be  greatest  when  the  moon's  distance  from  the 
earth  is  greatest,  and  the  sun's  distance  is  least  The  sun's  great- 
est apparent  semi-diameter  at  the  moon  is  16'  20".2.  Hence  CIM 
16'  22".5. 

In  the  triangle  IKM,  the  angle  CKM,  when  least,  is  14'  41".0; 
and  KM,  when  greatest,  is  249,307  miles. 

Then   sin.  CIM  :  sin.  CKM  : :  KM :  IM  =  223,552  miles. 

Hence  IE =476,815  miles. 

Then,  in  the  triangle  IEK, 

EK :  IE  : :  sin.  EIK  :  sin.  EKI  =  144°  58'  10". 

Hence  EKN=35°  1'  50". 

The  angle  KEI=EKN-EIK=34°  45'  28"=the  arc  KG. 

Hence  the  entire  arc  KL  =  69°  30'  56";  and  if  we  allow  69 \ 
miles  to  a  degree,  the  breadth  of  the  penumbra  is  4808  miles, 
nearly. 

310.  Velocity  of  the  moon's  shadow  over  tiie  earth. — The  moon  ad- 
vances eastward  among  the  stars  about  30'  per  hour  more  than 
the  sun ;  and  30'  of  the  moon's  orbit  is  about  2070  miles,  which 
therefore  we  may  consider  as  the  hourly  velocity  with  which  the 
moon's  shadow  passes  over  the  earth,  or  at  least  over  that  part  of 
it  on  which  the  shadow  falls  perpendicularly ;  in  every  other 
place  the  velocity  will  be  increased  in  the  ratio  of  the  sine  of  the 
angle  which  MV  makes  with  the  surface,  in  the  direction  of  its 


ECLIPSES  OF  THE  SUN.  169 

motion,  to  radius.  But  the  earth's  rotation  upon  its  axis  will  also 
affect  the  apparent  velocity  of  the  shadow,  and,  consequently,  the 
duration  of  the  eclipse  at  any  point  of  the  earth.  If  the  point  be 
moving  in  the  direction  of  the  shadow,  its  velocity  in  respect  to 
that  point  will  be  diminished,  and,  consequently,  the  time  in  which 
the  shadow  passes  over  that  point  will  be  increased ;  but  if  the 
point  be  moving  in  a  direction  contrary  to  that  of  the  shadow,  as 
may  happen  at  places  within  the  polar  circle,  the  relative  velocity 
of  the  shadow  will  be  increased,  and  the  time  diminished. 

•811.  Different  kinds  of  eclipses  of  the  sun. — A  partial  eclipse  of 
the  sun  is  one  in  which  a  part,  but  not  the  whole,  of  the  sun  is  ob- 
scured. A  total  eclipse  is  one  in  which  the  sun  is  entirely  ob- 
scured. It  must  occur  at  all  those  places  on  which  the  moon's 
shadow  falls.  A  central  eclipse  is  one  in  which  the  axis  of  the 
moon's  shadow,  or  the  axis  produced,  passes  through  a  given 
place.  An  annular  eclipse  is  one  in  which  a  part  of  the  sun's  disc 
is  seen  as  a  ring  surrounding  the  moon. 

The  apparent  discs  of  the  sun  and  moon,  though  nearly  equal, 
are  subject  to  small  varia-  Fig.  so. 

tions,  corresponding  to  their 
variations  of  distance,  in  con- 
sequence of  which  the  disc  of 
the  moon  is  sometimes  a  little 
greater,  and  sometimes  a  little 
less  than  that  of  the  sun.  If 
the  centres  of  the  sun  and  1 
moon  coincide,  and  the  disc 
of  the  moon  be  less  than  that 
of  the  sun,  the  moon  will  cov- 
er the  central  portion  of  the 
sun,  but  will  leave  uncovered 
around  it  a  regular  ring  or  an- 
nulus,  as  shown  in  Fig.  86.  This  is  called  an  annular  eclipse. 

312.  Duration  of  total  and  annular  eclipses. — The  greatest  value 
of  the  apparent  radius  of  the  moon,  as  seen  from  the  earth's  cen- 
tre, is  1006",  which  may  be  increased  by  the  moon's  elevation 
above  the  horizon,  Art.  215,  to  1024" ;  and  the  least  value  of  the 
radius  of  the  sun  is  945".  Their  difference  is  79".  The  greatest 


170 


ASTRONOMY. 


possible  duration  of  a  total  solar  eclipse  will  be  the  time  required 
for  the  centre  of  the  moon  to  gain  upon  that  of  the  sun  twice  79'v, 
or  158",  which  would  be  about  5m.  if  the  earth  did  not  rotate  upon 
an  axis;  but,  allowing  for  the  earth's  rotation,  the  greatest  possi- 
ble time  during  which  the  sun  can  be  totally  obscured  is  7m.  58s. 
This  will  be  the  duration  at  the  equator.  In  the  latitude  of  Paris, 
the  greatest  possible  duration  of  a  total  eclipse  is  6m.  10s. 

The  greatest  apparent  radius  of  the  sun  being  978",  and  the 
least  apparent  radius  of  the  moon  being  881",  the  greatest  possi- 
ble breadth  of  the  annul  us,  when  the  eclipse  is  central,  is  97". 
The  greatest  interval  during  which  the  eclipse  can  continue  an- 
nular is  the  time  required  for  the  centre  of  the  moon  to  gain  upon 
that  of  the  sun  twice  97",  or  19-i",  which  would  be  about  7m.  if 
the  earth  did  not  rotate;  but,  by  the  earth's  rotation,  this  quantity 
may  be  increased  to  12m.  24s.  at  the  equator.  In  the  latitude  of 
Paris,  the  greatest  possible  duration  of  an  annular  eclipse  is  9m. 
56s. 

Since  the  visual  directions  of  the  centres  of  the  sun  and  moon 
vary  with  the  position  of  the  observer  on  the  earth's  surface,  an 
eclipse  which  is  total  at  one  place  may  be  partial  at  another,  while 
at  other  places  no  eclipse  whatever  may  occur. 

Since  the  moon's  apparent  diameter  increases  as  her  elevation 
above  the  horizon  increases,  it  sometimes  happens,  when  the  ap- 
parent diameters  of  the  sun  and  moon  are  very  nearly  equal,  that 
the  apparent  diameter  of  the  moon,  when  near  the  horizon,  is  a 
little  less  than  that  of  the  sun,  but  becomes  a  little  greater  than 
that  of  the  sun  as  it  approaches  the  meridian ;  that  is,  an  eclipse 
which  is  annular  at  places  where  it  occurs  near  sunrise,  may  be 
total  at  places  where  it  occurs  near  midday. 

313.  To  compute  the  values  of  the  solar  ecliptic  limits. — No  eclipse 
of  the  sun  can  take  place  unless  some  part  of  the  globe  of  the 
moon  pass  within  the  lines  AC  and  BD,  which  touch  externally 


Fig.  87 


ECLIPSES  OF  THE   SUN.  171 

the  globes  of  the  sun  and  earth.  The  apparent  distance,  MES, 
of  the  moon's  centre  from  the  ecliptic  at  this  limit  is  equal  to 
AEF+ AES+FEM.  But  AEF^EFC-EAC.  Hence  MES= 
EFC-EAC-f  AES+FEM  =/_p+s'+s;  that  is,  the  sum  of  the 
apparent  semi-diameters  -of  the  sun  and  moon,  plus  the  difference 
of  their  horizontal  parallaxes.  Taking  the  greatest  and  least  val- 
ues of  these  quantities,  we  obtain 

the  greatest  value  of  MES=1°  34'  14", 
and  the  least  value =1°  24'  19". 

Computing  the  corresponding  distances  from  the  moon's  node, 
as  in  Art.  289,  we  find  that  if,  at  the  time  of  conjunction,  the  sun's 
distance  from  the  moon's  node  is  more  than  18°  20',  an  eclipse  is 
impossible ;  and  if  its  distance  from  the  node  is  less  than  15°  25', 
an  eclipse  is  inevitable.  Between  these  limits  an  eclipse  may  or 
may  not  occur,  according  to  the  magnitude  of  the  parallaxes  and 
apparent  diameters. 

Since,  then,  an  eclipse  can  only  take  place  within  a  few  degrees 
of  the  moon's  node,  and  the  sun  passes  the  two  nodes  of  the  moon 
at  opposite  seasons  of  the  year,  it  is  evident  that  if  an  eclipse  oc- 
curs in  January, one  or  more  eclipses  may  be  expected  in  July; 
but  no  eclipse,  either  of  the  sun  or  moon,  could  possibly  happen 
in  April  or  October  of  the  same  year. 

314.  lumber  of  eclipses  in  a  year. — There  may  be  seven  eclipses 
in  a  year,  and  can  not  be  less  than  two.  When  there  are  seven, 
five  of  them  are  of  the  sun  and  two  of  the  moon ;  when  there  are 
but  two,  they  are  both  of  the  sun. 

A  solar  eclipse  is  inevitable  if  conjunction  takes  place  within 
15°  25'  on  either  side  of  the  moon's  node,  comprehending  an  arc 
of  longitude  of  30°  50'.  Now,  during  a  synodic  revolution  of  the 
moon,  the  sun's  mean  motion  in  longitude  is  29°  6',  and  in  this 
time  the  moon's  nodes  move  backward  1°  31'.  Hence  the  sun's 
motion  with  reference  to  the  moon's  node,  in  one  lunation,  is  30° 
37',  which  is  less  than  30°  50'.  Hence  at  least  one  solar  eclipse 
must  occur  near  each  node  of  the  moon's  orbit,  and  therefore  there 
must  be  at  least  two  solar  eclipses  annually.  But  it  may  happen 
that  two  solar  eclipses  shall  occur  near  each  node,  and  also  one 
lunar  eclipse  ;  and  this  will  happen  if  opposition  takes  place  very 
near  the  moon's  node.  In  this  case  the  moon  will  be  almost  cen- 
trally eclipsed;  and  since  the  sun's  motion  in  reference  to  tho 


172  ASTRONOMY. 

node  during  half  a  lunation  is  only  15°  18',  it  is  evident  that,  both 
at  the  previous  and  following  new  moons,  the  sun  may  be  within 
the  ecliptic  limits  from  the  node,  and  may  therefore  be  eclipsed 
at  each  of  these  new  moons.  At  the  full  moon,  which  occurs  in 
a  little  less  than  six  months  after  the  former,  the  sun  will  be  near 
the  other  node  of  the  moon's  orbit.  Consequently,  there  must  be 
a  large  eclipse  of  the  moon,  and  there  may  be  an  eclipse  of  the 
sun  both  at  the  previous  and  following  new  moons.  At  the  new 
moon  which  occurs  five  and  a  half  lunations  after  this  latter  full 
moon,  and  therefore  a  little  before  the  close  of  the  year,  the  sun 
will  be  near  the  node  again,  and  must  therefore  be  eclipsed.  Thus 
there  may  be  two  eclipses  of  the  moon  and  five  of  the  sun  within 
a  period  of  twelve  months,  and  these  may  all  be  embraced  in  one 
calendar  year. 

315.  In  the  space  of  eighteen  years  there  are  usually  about  70 
eclipses,  29  of  the  moon  and  41  of  the  sun.     These  numbers  are 
nearly  in  the  ratio  of  two  to  three.    Nevertheless,  more  lunar  than 
solar  eclipses  are  visible  in  any  particular  place,  because  a  lunar 
eclipse  is  visible  to  an  entire  hemisphere,  while  a  solar  is  only 
visible  to  a  part 

The  last  eclipse  of  the  sun,  which  was  total  in  any  part  of 
the  United  States,  occurred  July  29,  1878,  and  was  total  in 
Colorado.  The  last  annular  eclipse  occurred  in  1875,  and  was 
annular  in  Massachusetts.  See  the  list  of  eclipses,  page  327. 

316.  Period  of  eclipses. — At  the  expiration  of  a  period  of  223 
lunations,  or  about  18  years  and  10  days,  eclipses,  both  of  the  sun 
and  moon,  return  again  in  nearly  the  same  order  as  during  that 
period. 

The  time  from  one  new  moon  to  another  is  29.53  days,  and, 
consequently,  223  lunations  include  6585.32  days. 

The  mean  period  in  which  the  sun  moves  from  one  of  the 
moon's  nodes  to  the  same  node  again  is  346.62  days,  because  the 
node  shifts  its  place  to  the  westward  19°  35'  per  annum.  This  pe- 
riod is  called  the  synodical  revolution  of  the  moon's  node.  Now 
19  synodical  revolutions  of  the  node  embrace  a  period  of  6585.78 
days.  Hence,  whatever  may  be  the  distance  of  the  sun  from  one 
of  the  moon's  nodes  at  any  new  or  full  moon,  he  must  at  the  end 
of  223  lunations,  be  nearly  at  the  same  distance  from  the  same 


ECLIPSES  OF  THE   SUX.  173 

node.  Hence,  after  a  period  of  6585.32  days  (which  is  18  years 
11 3-  days  when  there  are  four  bissextile  years  in  the  period,  or 
18  years  10^-  days  when  there  are  five),  eclipses  must  occur  again 
in  nearly  the  same  order  as  during  that  period.  This  period  waa 
known  to  the  Chaldaean  astronomers.  It  was  by  them  called  the 
Saros,  and  was  used  in  predicting  eclipses. 

On  page  327  is  given  a  list  of  eclipses,  which  will  illustrate  the 
period  of  the  Saros,  and  also  show  that  seven  eclipses  may  occur 
within  a  period  of  twelve  months. 

317.  Occultations. — When  the  moon  passes  between  the  earth 
and  a  star  or  planet,  she  must,  during  the  passage,  render  the  body 
invisible  to  some  parts  of  the  earth.     This  phenomenon  is  called 
an  occultation  of  the  star  or  planet     The  moon,  in  her  monthly 
course,  occults  every  star  which  is  included  in  a  zone  extending 
to  a  quarter  of  a  degree  on  each  side  of  the  apparent  path  of  her 
centre.     From  new  moon  to  full,  the  moon  moves  with  the  dark 
edge  foremost ;  and  from  full  moon  to  new,  it  moves  with  the 
bright  edge  foremost.     During  the  former  period,  stars  disappear 
at  the  dark  edge,  and  reappear  at  the  bright  edge;  while  during 
the  latter  period  they  disappear  at  the  bright  edge,  and  reappear 
at  the  dark  edge.     The  disappearance  of  a  star  at  the  dark  limb 
is  very  sudden  and  startling,  the  star  appearing  to  be  instantly 
annihilated  at  a  point  of  the  sky  where  nothing  is  seen  to  inter- 
fere with  it. 

318.  Darkness  attending  a  total  eclipse  of  the  sun. — During  a  total 
eclipse  of  the  sun,  the  darkness  is  generally  so  great  as  to  render 
the  brighter  stars  and  planets  visible.     Each  of  the  five  brighter 
planets  has  been  repeatedly  seen  during  the  total  obscuration  of 
the  sun ;  all  the  stars  of  the  first  magnitude  have  in  turn  been 
seen,  and,  on  some  occasions,  a  few  stars  of  the  second  magnitude 
have  been  detected.     During  a  total  eclipse,  the  degree  of  dark- 
ness is  therefore  somewhat  less  than  that  which  prevails  at  night 
in  presence  of  a  full  moon;  but  the  darkness  appears  much 
greater  than  this,  on  account  of  the  sudden  transition  from  day 
to  night. 

This  darkness,  however,  has  little  resemblance  to  the  usual 
darkness  of  the  night,  but  is  attended  by  an  unnatural  gloom, 
which  is  sometimes  tinged  with  green,  sometimes  red,  and  some- 


174  ASTRONOMY. 

times  a  yellowish-crimson.  The  color  of  the  sky  changes  from 
its  usual  azure  blue  to  a  livid  purple  or  violet  tint.  The  color  of 
surrounding  objects  becomes  yellowish,  or  of  a  light  olive  or 
greenish  tinge ;  and  the  figures  of  persons  assume  an  unearthly, 
cadaverous  aspect. 

319.  Moon  sometimes  visible  in  an  eclipse  of  the  sun. — During  a 
total  eclipse  of  the  sun,  the  moon's  surface  is  sometimes  faintly 
illumined  by  a  purplish-gray  light,  spreading  over  every  part  of 
the  disc,  so  that  the  light  of  the  disc  is  quite  noticeable  to  the 
naked  eye.     In  the  eclipse  of  May  3, 1733,  lunar  spots  were  dis- 
tinctly observed  by  Vassenius  at  Gottenberg.    This  effect  is  pro- 
duced by  the  sun's  light  reflected  from  the  earth  to  the  moon ;  for 
the  side  of  the  earth  which  at  such  times  is  presented  to  the  moon 
is  wholly  illumined  by  the  sun,  and  the  light  of  the  earth  is  about 
14  times  that  of  the  full  moon. 

320.  Bright  points  on  the  moon's  disc. — During  the  total  eclipse 
of  June  24,  1778,  about  a  minute  and  a  quarter  before  the  sun  be- 
gan to  emerge  from  behind  the  moon's  disc,  Ulloa  discovered, 
near  the  northwest  part  of  the  moon's  limb,  a  small  point  of  light, 
estimated  as  equal  to  a  star  of  the  fourth  magnitude.     This  point 
gradually  increased,  and  became  equal  to  a  star  of  the  second 
magnitude,  when  it  united  with  the  edge  of  the  sun,  which  at  that 
instant  emerged  from  behind  the  moon.     This  phenomenon  was 
doubtless  due  to  the  sun's  rays  shining  through  a  deep  valley  on 
the  moon's  limb,  and  the  long  continuance  of  this  light  was  due 
to  the  moon's  motion  being  nearly  parallel  to  that  portion  of  the 
sun's  circumference. 

A  similar  phenomenon  was  seen  by  M.Yalz,  of  Marseilles,  dur- 
ing the  eclipse  of  July  8, 1842. 

Again  the  same  phenomenon  was  seen  during  the  eclipse  of 
July  18, 1860,  in  Algeria,  by  two  French  observers,  one  with  the 
naked  eye,  and  the  other  with  a  telescope.  The  bright  point 
gradually  increased,  until  it  blended  with  the  light  of  the  sun's 
disc  as  it  emerged  from  behind  the  moon. 

During  the  eclipse  of  May  15, 1836,  about  25  seconds  before  the 
middle  of  the  eclipse,  Professor  Bessel,  with  the  Konigsberg  heli- 
ometer,  observed  a  faint  point  of  light  near  the  edge  of  the  moon's 
limb.  The  point  became  brighter,  and  other  similar  points  ap- 


ECLIPSES  OF  THE   SUN.  175 

peared  beside  it,  which  soon  united,  and  in  this  manner  rendered 
visible  the  whole  of  the  moon's  border  between  the  extremities 
of  the  sun's  cusps. 

Analogous  phenomena  have  been  observed  in  the  occultation 
of  stars  by  the  moon.  When  a  star  just  grazes  the  northern  or 
southern  limb  of  the  moon,  it  sometimes  disappears  behind  a 
lunar  mountain,  and  reappears  through  an  adjacent  valley,  to  dis- 
appear again  behind  the  next  mountain.  Several  such  disappear- 
ances and  reappearances  have  been  observed  within  an  interval 
of  a  few  minutes. 

321.  The  corona. — During  the  total  obscuration  of  the  sun,  the 
dark  body  of  the  moon  appears  surrounded  by  a  ring  of  light 
called  the  corona.     This  ring  is  of  variable  extent,  and  resembles 
the  "  glory"  with  which  painters  encircle  the  heads  of  saints.     It 
is  brightest  next  to  the  moon's  limb,  and  gradually  fades  to  a  dis- 
tance equal  to  one  third  of  her  diameter,  when  it  becomes  con- 
founded with  the  general  tint  of  the  heavens.     Sometimes  its 
breadth  is  nearly  equal  to  that  of  the  moon's  diameter.     The 
corona  generally  begins  5  or  6  seconds  before  the  total  obscura- 
tion of  the  sun,  and  continues  a  few  seconds  after  the  sun's  reap- 
pearance.   Sometimes  the  corona  is  distinctly  seen  at  places  where 
the  eclipse  is  not  quite  total. 

The  color  of  the  corona  has  been  variously  described.  Some- 
times it  is  compared  to  the  color  of  tarnished  silver.  Sometimes 
it  is  described  as  of  a  pearl  white ;  sometimes  of  a  pale  yellow ; 
sometimes  of  a  golden  hue ;  sometimes  peach-colored,  and  some- 
times reddish. 

The  intensity  of  the  light  of  the  corona  is  sometimes  such  that 
the  eye  is  scarcely  able  to  support  it ;  but  generally  it  is  described 
as  precisely  similar  to  that  of  the  moon. 

The  corona  generally  presents  somewhat  of  a  radiated  appear- 
ance. Sometimes  these  rays  are  very  strongly  marked ;  and  long 
beams  have  occasionally  been  traced  to  a  distance  of  3°  or  4°  from 
the  moon's  limb. 

322.  Cause  of  the  corona. — Some  have  maintained  that  this  co- 
rona is  caused  by  the  diffraction  of  the  sun's  light  in  its  passage 
near  the  edge  of  the  moon.    But  the  diffracted  light,  surrounding 
an  opaque  circular  disc,  consists  of  concentric  rings  exhibiting  a 


176  ASTRONOMY. 

regular  succession  of  colors — pale  blue,  yellow,  and  red.  If  the 
corona  seen  in  solar  eclipses  were  due  to  diffraction,  it  ought  to 
exhibit  a  series  of  concentric  colored  rings,  like  those  seen  sur- 
rounding the  moon  when  obscured  by  a  thin  haze.  Such  is  not 
the  appearance  actually  observed.  It  is  more  probable  that  this 
corona  is  due  to  an  atmosphere  surrounding  the  sun,  extending  to 
a  height  of  several  thousand  miles  above  its  disc,  and  reflecting  a 
portion  of  the  sun's  light. 

The  radiated  appearance  of  the  corona  is  probably  analogous 
to  the  rays  which  are  frequently  seen  in  the  western  sky  after 
sunset,  and  which  are  caused  by  the  shadows  of  clouds  situated 
near,  or  perhaps  below  our  visible  horizon.  In  like  manner,  the 
clouds  which  float  in  the  solar  atmosphere  intercept  a  portion  of 
the  light  of  the  sun's  disc,  and  the  space  behind  them  is  less 
bright  than  that  portion  of  space  which  is  illumined  by  the  un- 
obstructed rays  of  the  sun. 

V323.  Baily  s  beads. — When,  in  the  progress  of  the  eclipse,  the 
sun's  disc  has  been  reduced  to  a  thin  crescent,  this  crescent  often 
Fi  8S  appears  as  a  band  of  brilliant  points,  sep- 

arated by  dark  spaces,  giving  it  the  ap- 
pearance of  a  string  of  brilliant  beads. 
The  same  peculiarity  is  noticed  in  an- 
nular eclipses  a  few  seconds  previous  to 
the  formation,  and  again  a  few  seconds 
previous  to  the  rupture,  of  the  annul  us. 
This  phenomenon  was  first  clearly  de- 
scribed by  Sir  Francis  Baily  on  occasion 
of  the  annular  eclipse  of  May  15, 1836, 
and  it  has  hence  acquired  the  name  of  Baily 's  leads.  This  ap- 
pearance is  generally  ascribed  to  the  inequalities  of  the  moon's 
surface.  The  outline  of  the  moon's  disc  is  not  a  perfect  circle, 
but  is  full  of  notches;  and  these  inequalities  are  easily  seen  when 
the  moon's  disc  is  projected  upon  that  of  the  sun.  Just  before 
the  commencement  of  the  total  eclipse,  the  tops  of  the  lunar 
mountains  extend  to  the  edge  of  the  sun's  disc,  but  still  permit 
the  sun's  light  to  glimmer  through  the  hollows  between  the 
mountain  ridges. 

These  appearances  are  materially  modified  by  the  color  of  the 
glass  through  which  the  observations  are  made.  They  are  most 


ECLIPSES   OF  THE   SUN.  177 

conspicuous  through  a  red  glass,  and  through  certain  colored 
glasses  are  scarcely  noticed  at  all.  This  peculiarity  is  probably 
due  to  the  unequal  penetrating  power  of  the  differently  colored 
rays  of  the  sun.  The  red  rays  of  the  sun  are  less  readily  ab- 
sorbed than  any  other  rays  of  the  spectrum ;  and  a  glass  which 
transmits  only  the  red  rays  will  allow  the  sun's  light  to  appear 
through  minute  crevices  in  the  edge  of  the  moon,  when  rays  of 
any  other  color  would  be  entirely  absorbed  by  the  colored  glass 
through  which  the  observation  is  made. 

324.  Flame-like  protuberances.  —  Immediately  after  the  com- 
mencement of  the  total  obscuration,  red  protuberances,  resem- 
bling flames,  may  be  seen  to  issue  from  behind  the  moon's  disc. 
These  appearances  were  noticed  in  the  eclipse  of  May  3, 1733, 
and  they  have  been  re-observed  during  every  total  solar  eclipse 
which  has  taken  place  since  that  time.  They  did  not,  however, 
attract  much  attention  before  the  eclipse  of  July  8, 1842,  when 
they  were  carefully  observed  and  delineated  in  accurate  diagrams. 
They  were  again  made  the  subject  of  special  study  in  the  eclipse 
of  July  28, 1851,  and  also  in  that  of  July  18, 1860. 

The  forms  of  these  protuberances  are  very  various,  and  some 
of  them  quite  peculiar.  Many  of  them  are  nearly  conical,  the 
height  being  frequently  greater  than  the  breadth  of  the  base. 
Others  resemble  the  tops  of  a  very  irregular  range  of  hills  stretch- 
ing continuously  along  one  sixth  of  the  moon's  circumference. 
Some  of  these  protuberances  reach  to  a  vast  height,  and  show  re- 
markable curvature.  One  has  been  compared  to  a  sickle ;  a  sec- 
ond to  a  Turkish  cimeter;  a  third  to  a  boomerang,  with  one  ex- 
tremity extending  off  horizontally  far  beyond  the  support  of  the 
base ;  while  a  fourth  was  of  a  circular  form,  entirely  detached 
from  the  moon's  limb  by  a  space  nearly  equal  to  its  own  breadth. 

The  size  of  these  protuberances  is  very  various.  Some  have 
been  estimated  to  have  an  apparent  height  of  3',  which  would  im- 
ply an  absolute  height  of  80,000  miles ;  while  others  have  every 
intermediate  elevation  down  to  the  smallest  visible  object. 

The  colors  of  these  protuberances  have  been  variously  de- 
scribed. Some  have  been  called  simply  reddish,  while  others 
have  been  characterized  as  rose-red,  purple,  or  scarlet;  and  a  few 
have  been  represented  as  nearly  white. 

During  the  solar  eclipses  of  1842, 1851,  and  18603  the  largest 
M 


178  ASTRONOMY. 

of  these  protuberances  were  seen  by  the  unassisted  eye.  In  1860, 
some  of  them  were  observed  several  seconds  before  the  total  ob- 
scuration ;  and  in  1842,  as  well  as  in  1851,  some  of  them  remained 
visible  from  5s.  to  7s.  after  the  sun's  emersion. 

325.  These  protuberances  emanate  from  the  sun. — These  protuber- 
ances emanate  from  the  disc  of  the  sun,  and  not  from  that  of  the 
moon.     This  is  proved  by  the  following  observations  made  in 
1851.     The  protuberances  seen  near  the  eastern  limb  decrease  in 
dimensions  from  the  commencement  of  the  total  eclipse  to  its 
close,  while  those  near  the  western  limb  increase  from  the  com- 
mencement to  the  close ;  indicating  that  the  moon  covers  more 
and  more  the  protuberances  on  the  eastern  side  of  the  sun's  disc, 
and  gradually  exposes  a  larger  and  larger  portion  of  the  protuber- 
ances on  the  western  side.    Again,  during  the  eclipse  of  1860,  the 
astronomers  who  went  to  Spain  to  observe  the  eclipse  obtained 
two  excellent  photographs,  in  which  these  flame-like  protuber- 
ances were  faithfully  copied ;  and  it  was  found  that  the  protuber- 
ances retained  a  fixed  position  with  reference  to  the  sun  as  the 
moon  glided  before  it;  and  they  did  not  change  their  form, ex- 
cept as  the  moon,  by  passing  over  them,  shut  them  off  on  the  east- 
ern side,  while  fresh  ones  became  visible  on  the  western  side.    See 
Plate  III. 

326.  Nature  of  these  protuberances. — That  these  protuberances 
are  not  solid  bodies  like  mountains  is  proved  by  their  peculiar 
forms,  the  tops  frequently  extending  horizontally  far  beyond  the 
support  of  the  base ;  and  they  sometimes  appear  entirely  detached 
from  the  sun's  disc  without  any  visible  support. 

The  same  argument  proves  that  they  are  not  liquid  bodies; 
and  hence  we  must  conclude  that  they  are  gaseous,  or  are  sus- 
tained in  a  gaseous  medium. 

These  flame-like  emanations  seem  to  be  analogous  to  the  clouds 
which  float  at  great  elevations  in  our  own  atmosphere ;  and  we 
are  naturally  led  to  infer  that  the  sun  is  surrounded  by  a  trans- 
parent atmosphere,  rising  to  a  height  exceeding  one  tenth  of  his 
diameter;  and  in  this  atmosphere  there  are  frequently  found 
cloudy  masses  of  extreme  tenuity  floating  at  various  elevation?, 
and  sometimes  rising  to  the  height  of  80,000  miles  above  the  lu- 
minous surface  of  the  sun. 


METHODS   OF   FINDING  THE   LONGITUDE.  179 


CHAPTER  XII. 

DIFFERENT  METHODS  OF  FINDING  THE  LONGITUDE   OF  A  PLACE. 

327.  Difference  of  time  under  different  meridians. — The  sun,  in 
his  apparent  diurnal  motion  from  east  to  west,  passes  successively 
over  the  meridians  of  different  places ;  and  noon  occurs  later  and 
later  as  we  travel  westward  from  any  given  meridian.     If  we 
start  from  the  meridian  of  Greenwich,  then  the  sun  will  cross  the 
meridian  of  a  place  15°  west  of  Greenwich  one  hour  later  than  it 
crosses  the  Greenwich  meridian — that  is,  at  one  o'clock,  Green- 
wich time.    A  difference  of  longitude  of  15°  corresponds  to  a  dif- 
ference of  one  hour  in  local  times.     In  order,  then,  to  determine 
the  longitude  of  any  place  from  Greenwich,  we  must  accurately 
determine  the  local  time,  and  compare  this  with  the  correspond- 
ing Greenwich  time. 

328.  Method  of  chronometers. — Let  a  chronometer  which  keeps 
accurate  time  be  carefully  adjusted  to  the  time  of  some  place 
whose  longitude  is  known — for  example,  Greenwich  Observatory^. 
Then  let  the  chronometer  be  carried  to  a  place  whose  longitude 
is  required,  and  compared  with  the  correct  time  reckoned  at  that 
place.     The  difference  between  this  time  and  that  shown  by  the 
chronometer  will  be  the  difference  of  longitude  between  the  given 
place  and  Greenwich. 

It  is  not  necessary  that  the  chronometer  should  be  so  regulated 
as  neither  to  gain  nor  lose  time.  This  would  be  difficult,  if  not 
impracticable.  It  is  only  necessary  that  its  rate  should  be  well 
ascertained,  since  an  allowance  can  then  be  made  for  its  gain  or 
loss  during  the  time  of  its  transportation  from  one  place  to  the 
other. 

The  manufacture  of  chronometers  has  attained  to  such  a  degree 
of  perfection  that  this  method  of  determining  difference  of  longi- 
tude, especially  of  stations  not  very  remote  from  each  other,  is 
one  of  the  best  methods  known.  The  most  serious  difficulty  in 


180  ASTRONOMY. 

the  application  of  the  method  consists  in  determining  the  rate  of 
the  chronometer  during  the  journey;  for  chronometers  generally 
have  a  different  rate,  when  transported  from  place  to  place,  from 
that  which  they  maintain  in  an  observatory.  For  this  reason, 
when  great  accuracy  is  required,  it  is  customary  to  employ  a 
large  number  of  chronometers  as  checks  upon  each  other;  and 
the  chronometers  are  transported  back  and  forth  a  considerable 
number  of  times. 

This  is  the  method  by  which  the  mariner  commonly  determ- 
ines his  position  at  sea.  Every  day,  when  practicable,  he  meas- 
ures the  sun's  altitude  at  noon,  and  hence  determines  his  latitude. 
About  three  hours  before  or  after  noon  he  measures  the  sun's  al- 
titude again,  and  from  this  he  computes  his  local  time  by  Art.  145. 
The  chronometer  which  he  carries  with  him  shows  him  the  true 
time  at  Greenwich,  and  the  difference  between  the  two  times  is 
his  longitude  from  Greenwich. 

329.  By  eclipses  of  the  moon. — An  eclipse  of  the  moon  is  seen  at 
the  same  instant  of  absolute  time  in  all  parts  of  the  earth  where 
the  eclipse  is  visible.     Therefore,  if  at  two  distant  places  the  times 
of  the  beginning  of  the  eclipse  are  carefully  observed,  the  differ- 
ence of  these  times  will  give  the  difference  of  longitude  between 
the  places  of  observation ;  but,  on  account  of  the  gradually  in- 
creasing darkness  of  the  penumbra,  it  is  impossible  to  determine 
the  precise  instant  when  the  eclipse  begins,  and  therefore  this 
method  is  of  no  value  except  under  circumstances  which  preclude 
the  use  of  better  methods. 

330.  By  (he  eclipses  of  Jupiter's  satellites. — The  moons  of  Jupiter 
are  eclipsed  by  passing  into  the  shadow  of  Jupiter  in  the  same 
manner  as  our  moon  is  eclipsed  by  passing  into  the  shadow  of  the 
earth.     These  eclipses  begin  at  the  same  instant  of  absolute  time 
for  all  places  at  which  they  are  visible.     If,  then,  the  times  of  the 
beginning  of  an  eclipse  be  accurately  observed  at  two  different 
places,  the  difference  of  these  times  will  be  the  difference  of  longi- 
tude of  the  places.     Since,  however,  the  light  of  a  satellite  dimin- 
ishes gradually  while  entering  the  shadow,  and  increases  gradu- 
ally on  leaving  it,  the  observed  time  of  beginning  or  ending  of  the 
eclipse  must  depend  on  the  power  of  the  telescope  used,  and  also 
upon  the  eye  of  the  observer.     This  method,  therefore,  is  of  nc 


METHODS   OF   FINDING  THE   LONGITUDE.  181 

value  at  fixed  observatories,  "where  better  methods  are  always 
available. 

331,  By  an  eclipse  of  the  sun  or  the  occupation  of  a  star. — The 
times  of  the  beginning  and  end  of  an  eclipse  of  the  sun,  or  of  the 
occultation  of  a  star  or  planet  at  any  place,  depend  on  the  posi- 
tion of  the  place.     We  can  not,  therefore,  use  a  solar  eclipse  as 
an  instantaneous  signal  for  comparing  directly  the  local  times  at 
two  stations;  but  we  may  deduce  by  computation  from  the  ob- 
served beginning  and  end  of  an  eclipse,  the  time  of  true  conjunc- 
tion of  the  sun  and  moon — that  is,  the  time  of  conjunction  as  seen 
from  the  centre  of  the  earth ;  and  this  is  a  phenomenon  which 
happens  at  the  same  absolute  instant  for  every  observer  on  the 
earth's  surface.     If  the  eclipse  has  been  observed  under  two  dif- 
ferent meridians,  we  may  determine  the  instant  of  true  conjunc- 
tion from  the  observations  at  each  station ;  and  since  the  abso- 
lute instant  of  this  phenomenon  is  the  same  for  both  places,  the 
difference  of  the  results  thus  obtained  is  the  difference  of  longi- 
tude of  the  two  stations.     This  is  one  of  the  most  accurate  meth- 
ods known  to  astronomers  for  determining  the  difference  of  longi- 
tude of  two  stations  remote  from  each  other.     This  is  especially 
true  when  the  moon  crosses  a  cluster  containing  a  large  number 
of  stars,  as  the  Pleiades. 

332.  By  moon  culminating  stars. — Certain  stars  situated  near  the 
moon's  path,  and  passing  the  meridian  at  short  intervals  before 
or  after  the  moon,  are  called  moon  culminating  stars.     The  moon's 
motion  in  right  ascension  is  very  rapid,  amounting  to  about  half 
a  degree,  or  two  minutes  in  time,  during  a  sidereal  hour — that  is, 
during  the  interval  that  elapses  from  the  time  a  star  is  on  the  me- 
ridian of  any  place,  till  it  is  on  the  meridian  of  a  place  whose 
longitude  is  15°  west  of  the  former.    Hence  the  intervals  between 
the  passages  of  the  moon  and  a  star  over  the  meridians  of  two 
places  differing  an  hour  in  longitude  must  differ  about  two  min- 
utes; and  for  other  differences  of  longitude  there  must  be  a  pro- 
portional difference  in  the  intervals.     Hence,  if  the  intervals  be- 
tween the  passages  of  the  moon  and  a  star  over  the  meridians  of 
two  places  be  accurately  observed,  the  difference  of  their  longi- 
tude may  be  found  by  means  of  the  moon's  hourly  variation  in 
right  ascension. 


132  ASTRONOMY. 

The  chief  disadvantage  of  this  method  consists  in  this  circum- 
stance, that  an  error  in  the  observed  increase  of  right  ascension 
will  produce  an  error  nearly  30  times  as  great  in  the  computed 
longitude.  Hence,  to  obtain  a  satisfactory  result  by  this  method 
requires  a  series  of  observations  made  with  the  utmost  care,  and 
continued  through  a  long  period  of  time. 

333.  By  lunar  distances. — The  Nautical  Almanac  furnishes  for 
each  day  the  distance  of  the  moon  from  the  sun,  the  larger  plan- 
ets, and  several  stars  situated  near  the  moon's  path.  These  dis- 
tances are  given  for  Greenwich  time,  and  are  such  as  they  would 
appear  to  a  spectator  placed  at  the  centre  of  the  earth.  A  mari- 
ner on  the  ocean  measures  with  a  sextant  the  distance  from  the 
moon  to  one  of  the  stars  mentioned  in  the  Almanac.  He  corrects 
this  distance  for  refraction  and  parallax,  and  thus  obtains  the  true 
lunar  distance  as  it  would  be  seen  at  the  centre  of  the  earth.  By 
other  observations,  he  knows  the  local  time  at  which  this  distance 
was  measured,  and,  by  referring  to  the  Nautical  Almanac,  he  finds 
the  Greenwich  time  at  which  the  lunar  distance  was  the  same. 
The  difference  between  the  local  time  and  the  Greenwich  time 
represents  the  longitude  of  the  place  of  observation  from  Green- 
wich. This  method  of  finding  the  longitude  may  be  practiced  at 
sea,  and  in  long  voyages  should  always  be  resorted  to  as  a  check 
upon  the  method  by  chronometers. 

33-i.  By  the  electric  telegraph. — The  difference  of  the  local  times 
of  two  places  may  be  determined  by  means  of  any  signal  which 
can  be  seen  or  heard  at  both  places  at  the  same  instant.  When 
the  places  are  not  very  distant,  the  explosion  of  a  rocket,  or  the 
flash  of  gunpowder,  or  the  flight  of  a  shooting  star  may  serve 
this  purpose. 

The  electric  telegraph  affords  the  means  of  transmitting  signals 
to  a  distance  of  a  thousand  miles  or  more  with  scarcely  any  ap- 
preciable loss  of  time.  Suppose  that  there  are  two  observatories 
at  a  considerable  distance  from  each  other,  and  that  each  is  pro- 
vided with  a  good  clock,  and  with  a  transit  instrument  for  determ- 
ining its  error;  then,  if  they  are  connected  by  a  telegraph  wire, 
they  have  the  means  of  transmitting  signals  at  pleasure  from  either 
observatory  to  the  other  for  the  purpose  of  comparing  their  local 
times.  For  convenience,  we  \\\\\  call  the  most  eastern  station  E, 


METHODS   OF   FINDING  THE  LONGITUDE.  183 

and  the  western  W.     The  following  is  one  mode  of  comparing 
their  local  times. 

335.  Mode  of  comparing  the  local  times. — The  plan  of  operations 
having  been  previously  agreed  upon,  the  astronomer  at  E  strikes 
the  key  of  his  register,  and  makes  a  record  of  the  time  according 
to  his  observatory  clock.     Simultaneously  with  this  signal  at  E, 
the  armature  of  the  magnet  at  W  is  moved,  producing  a  click  like 
the  ticking  of  a  watch.     The  astronomer  at  W  hears  the  sound, 
and  notes  the  instant  by  his  clock.     The  difference  between  the 
time  recorded  at  E  and  that  at  W  is  the  difference  between  the 
two  clocks.     A  single  comparison  of  this  kind  will  furnish  the 
difference  of  longitude  to  the  nearest  second ;  but  to  obtain  the 
fraction  of  a  second  with  the  greatest  precision  requires  many 
repetitions,  and  this  is  accomplished  as  follows : 

At  the  commencement  of  the  minute  by  his  clock,  the  astron- 
omer at  E  strikes  his  signal  key,  and  the  time  of  the  signal  is  re- 
corded both  at  E  and  "W.  At  the  close  of  10  seconds  the  signal 
is  repeated,  and  the  observation  is  recorded  at  both  stations.  The 
same  thing  is  done  at  the  end  of  20  seconds,  of  30  seconds,  and  so 
on  to  20  repetitions.  The  astronomer  at  "W  then  transmits  a  se- 
ries of  signals  in  the  same  manner,  and  the  times  are  recorded  at 
both  stations. 

336.  The  velocity  of  the  electric  fluid. — This  double  set  of  signals 
not  only  furnishes  an  accurate  comparison  of  the  two  clocks,  but 
also  enables  us  to  measure  the  velocity  of  the  electric  fluid.     If 
the  fluid  requires  no  time  for  its  transmission,  then  the  apparent 
difference  between  the  two  clocks  will  be  the  same,  whether  we 
determine  it  by  signals  transmitted  from  E  to  W,  or  from  W  to 
E.     But  if  the  fluid  requires  time  for  its  transmission,  these  re- 
sults will  differ.     Suppose  the  true  difference  of  longitude  be- 
tween the  places  is  one  hour,  and  that  it  requires  one  second  for 
a  signal  to  be  transmitted  from  E  to  W.     Then,  if  at  10  o'clock 
a  signal  be  made  and  recorded  at  E,  it  will  be  a  second  before  the 
signal  is  heard  and  recorded  at  "W — that  is,  the  time  recorded  at 
W  will  be  9  hours  and  1  second ;  and  the  apparent  difference  be- 
tween the  two  clocks  will  be  59  minutes  and  59  seconds.     But  if 
a  signal  be  made  at  "W  at  nine  o'clock,  it  will  be  heard  at  E  at  10 
hours  and  1  second ;  and  the  apparent  difference  between  the  two 


184  ASTRONOMY. 

clocks  will  be  1  hour  and  1  second.  Thus  the  differences  between 
the  two  clocks,  as  derived  from  the  two  methods  of  comparison, 
diifer  by  twice  the  time  required  for  the  transmission  of  a  signal 
from  E  to  W.  Numerous  observations,  made  on  the  longest  lines 
and  with  the  greatest  care,  have  shown  that  the  velocity  of  the 
electric  fluid  upon  the  telegraph  wires  is  about  16,000  miles  per 
second.  The  mean  of  the  results  obtained  by  signals  transmit- 
ted in  both  directions,  gives  the  true  difference  between  the  two 
clocks,  independent  of  the  time  required  in  the  transmission  of 
signals. 

337.  How  the  clock  may  break  the  electric  circuit. — The  most  ac- 
curate method  of  determining  difference  of  longitude  consists  in 
employing  one  of  the  clocks  to  break  the  electric  circuit  each  sec- 
ond. This  may  be  accomplished  in  the  following  manner :  Near 
the  lower  extremity  of  the  pendulum  place  a  small  metallic  cup 
containing  a  globule  of  mercury,  so  that  once  in  every  vibration 
the  pointer  at  the  end  of  the  pendulum  may  pass  through  the 
mercury.  A  wire  from  one  pole  of  the  battery  is  connected  with 
the  supports  of  the  pendulum,  and  another  wire  from  the  other 
pole  of  the  battery  connects  with  the  cup  of  mercury.  When  the 
pointer  is  in  the  mercury,  the  electric  circuit  will  be  complete 
through  the  pendulum ;  but  as  soon  as  it  passes  out  of  the  mer- 
cury, the  circuit  will  be  broken. 

When  the  connections  are  properly  made,  there  will  be  heard 
a  click  of  the  magnet  at  each  station,  simultaneously  with  the 
beats  of  the  electric  clock.  If  each  station  be  furnished  with  an 
ordinary  Morse  register,  there  will  be  traced  upon  the  paper  a  se- 
ries of  lines,  of  equal  length,  separated  by  short  intervals,  thus: 

The  mode  of  using  the  register  for  marking  the  date  of  any 
event  is  to  strike  the  key  of  the  register  at  the  required  instant, 
when  an  interruption  will  be  made  in  one  of  the  lines  of  the  grad- 
uated scale ;  and  its  position  will  indicate  not  only  the  second, 
but  the  fraction  of  a  second  at  which  the  signal  was  made. 

We  now  employ  the  same  electric  circuit  for  telegraphing  tran- 
sits of  stars.  A  list  of  stars  having  been  selected  beforehand,  and 
furnished  to  each  observer,  the  astronomer  at  E  points  his  transit 
telescope  upon  one  of  the  stars  as  it  is  passing  his  meridian,  and 
strikes  the  key  of  his  register  at  the  instant  the  star  passes  sue- 


THE  TIDES.  185 

cessively  each  wire  of  his  transit,  and  the  dates  are  recorded,  not 
only  upon  his  own  register,  but  also  upon  that  at  "W.  When  the 
same  star  passes  over  the  meridian  of  W,  the  observer  there  goes 
through  the  same  operations,  and  his  observations  are  printed 
upon  both  registers.  These  observations  furnish  the  difference 
of  longitude  of  the  two  stations,  independently  of  the  tabular  place 
of  the  star  employed,  and  also  independently  of  the  absolute  er- 
ror of  the  clock. 


CHAPTER  XIII. 

THE  TIDES. 

338.  Definitions. — The  alternate  rise  and  fall  of  the  surface  of 
the  sea  twice  in  the  course  of  a  lunar  day,  or  of  24h.  51m.  of  mean 
solar  time,  is  the  phenomenon  known  by  the  name  of  the  tides. 
When  the  water  is  rising  it  is  said  to  be  flood  tide,  and  when  it  is 
falling,  ebb  tide.    When  the  water  is  at  its  greatest  height  it  is  said 
to  be  high  water,  and  when  at  its  least  height,  low  water. 

339.  Spring  and  neap  tides. — The  time  from  one  high  water  to 
the  next  is,  at  a  mean,  12h.  25m.  24s.     Near  the  time  of  new  and 
full  moon  the  tide  is  the  highest,  and  the  interval  between  the 
consecutive  tides  is  the  least,  viz.,  12h.  19m.     Near  the  quadra- 
tures, or  when  the  moon  is  90°  distarft  from  the  sun,  the  tides  are 
the  least,  and  the  interval  between  them  is  the  greatest,  viz.,  12h. 
30m.     The  former  are  called  the  spring  tides,  and  the  latter  the 
neap  tides.     At  New  York  the  average  height  of  the  spring  tides 
is  5  A  feet,  and  of  the  neap  tides  3 A  feet. 

340.  The  establishment  of  a  port. — The  time  of  high  water  is 
mostly  regulated  by  the  moon  ;  and  for  any  given  place,  the  hour 
of  high  water  is  always  nearly  at  the  same  distance  from  that  of 
the  moon's  passage  over  the  meridian.     The  mean  interval  be- 
tween the  moon's  passage  over  the  meridian,  and  high  water  at 
any  port  on  the  days  of  new  and  full  moon,  is  called  the  establish- 
ment of  the  port.     The  mean  interval  at  New  York  is  8h.  13m., 
and  the  difference  between  the  greatest  and  the  least  interval  oc- 
curring in  different  parts  of  the  month  is  43  minutes. 


186  ASTRONOMY. 

3-il.  Tides  at  perigee  and  apogee. — The  height  of  the  tide  is  af- 
fected by  the  distance  of  the  moon  from  the  earth,  being  highest 
near  the  time  when  the  moon  is  in  perigee,  and  lowest  near  the 
time  when  she  is  in  apogee.  When  the  moon  is  in  perigee,  at  or 
near  the  time  of  a  new  or  full  moon,  unusually  high  tides  oc- 


342.  Cause  oftiie  tides. — The  facts  just  stated  indicate  that  the 
moon  has  some  agency  in  producing  the  tides.  The  tides,  how- 
ever, are  not  due  to  the  simple  attraction  of  the  moon  upon  the 
earth,  but  to  the  difference  of  its  attraction  on  the  opposite  sides 


Fig.  89. 


of  the  earth.  Let  ACEG  represent  the  earth,  and  let  us  suppose 
its  entire  surface  to  be  covered  with  water ;  also,  let  M  be  the 
place  of  the  moon.  The  different  parts  of  the  earth's  surface  are 
at  unequal  distances  from  the  moon.  Hence  the  attraction  which 
the  moon  exerts  at  A  is  greater  than  that  which  it  exerts  at  B 
and  II,  and  still  greater  than  that  which  it  exerts  at  C  and  G; 
while  the  attraction  which  it  exerts  at  E  is  least  of  all.  The  at- 
traction which  the  moon  exerts  upon  the  mass  of  water  imme- 
diately under  it,  near  the  point  Z,  is  greater  than  that  which  it 
exerts  upon  the  solid  mass  of  the  globe.  The  water  will  there- 
fore heap  itself  up  over  A,  forming  a  convex  protuberance — that 
is,  high  water  will  take  place  immediately  under  the  moon.  The 
water  which  thus  collects  at  A  will  flow  from  the  regions  C  and 
G,  where  the  quantity  of  water  must  therefore  be  diminished — 
that  is,  there  will  be  low  water  at  C  and  G. 

The  water  at  N  is  less  attracted  than  the  solid  mass  of  the  earth. 
The  solid  mass  of  the  earth  will  therefore  recede  from  the  waters 
at  N,  leaving  the  water  behind,  which  will  thus  be  heaped  up  at 
N,  forming  a  convex  protuberance,  or  high  water,  similar  to  that 
at  Z.  The  sea  is  therefore  drawn  out  into  an  ellipsoidal  form, 
having  its  major  axis  directed  toward  the  moon. 


THE   TIDES. 


187 


343.  Effect  of  the  surts  attraction. — The  attraction  of  the  sun 
produces  effects  similar  to  those  of  the  moon,  but  less  powerful  in 
raising  a  tide,  because  the  inequality  of  the  sun's  attraction  on  dif- 
ferent parts  of  the  earth  is  very  small.    It  has  been  computed 
that  the  tidal  wave  due  to  the  action  of  the  moon  is  about  double 
that  which  is  due  to  the  sun. 

There  is,  therefore,  a  solar  as  well  as  a  lunar  tide  wave,  the  lat- 
ter greater  than  the  former,  and  each  following  the  luminary  from 
which  it  takes  its  name.  When  the  sun  and  moon  are  both  on 
the  same  side  of  the  earth,  or  on  opposite  sides,  that  is,  when  it 
is  new  or  full  moon,  their  effects  in  producing  tides  are  combined, 
and  the  result  is  an  unusually  high  tide,  called  spring  tide. 

When  the  moon  is  in  quadrature,  the  action  of  the  sun  tends 
to  produce  low  water  where  that  of  the  moon  produces  high  wa- 
ter, and  the  result  is  an  unusually  small  tide,  called  neap  tide. 

344.  Effect  of  the  moon's  declination  on  the  tides. — The  height  of 
the  tide  at  a  given  place  is  influenced  by  the  declination  of  the 
moon.     When  the  moon  has  no  declination,  the  highest  tides 
should  occur  along  the  equator;  and  the  heights  should  diminish 
from  thence  toward  the  north  and  south ;  but  the  two  daily  tides 
at  any  place  should  have  the  same  height.     When  the  moon  has 
north  declination,  as  shown  in  Fig.  90,  the  highest  tides  on  the 
side  of  the  earth  next  the  moon  will  be  at  places  having  a  cor- 

Fir.  00. 


A,'' 


responding  north  latitude,  as  at  B,  and  on  the  opposite  side  at 
those  which  have  an  equal  south  latitude.  And  of  the  two  daily 
tides  at  any  place,  that  which  occurs  when  the  moon  is  nearest 
the  zenith  should  be  the  greatest.  Hence,  when  the  moon's  dec- 


138  ASTRONOMY. 

lination  is  north,  the  height  of  the  tide  at  a  place  in  north  lati- 
tude should  be  greater  when  the  moon  is  above  the  horizon  than 
when  she  is  below  it.  At  the  same  time,  places  south  of  the 
equator  have  the  highest  tides  when  the  moon  is  below  the  hori- 
zon, and  the  least  when  she  is  above  it.  This  is  called  the  diur- 
nal inequality,  because  its  cycle  is  one  day ;  but  it  varies  greatly 
in  amount  at  different  places. 

The  great  wave  which  constitutes  the  tide  is  to  be  considered 
as  an  undulation  of  the  waters  of  the  ocean,  in  which  (except 
when  it  passes  over  shallows  or  approaches  the  shores)  there  is 
little  or  no  progressive  motion  of  the  water. 

345.  Why  the  phenomena  of  the  tides  are  so  complicated. — The 
actual  phenomena  of  the  tides  are  far  more  complicated  than  they 
would  be  if  the  earth  were  entirely  covered  with  an  ocean  of  great 
depth.  The  water  covers  less  than  three  quarters  of  the  earth's 
surface,  and  a  considerable  part  of  this  water  is  less  than  a  mile  in 
depth.  Two  great 'continents  extend  from  near  the  north  pole  to 
a  great  distance  south  of  the  equator,  thus  interrupting  the  regu- 
lar progress  of  the  tidal  wave  around  the  globe.  In  the  northern 
hemisphere,  the  waters  of  the  Atlantic  can  communicate  with 
those  of  the  Pacific  only  by  Behring's  Strait,  a  channel  36  mile? 
in  breadth,  which  effectually  prevents  the  transmission  of  any 
considerable  wave  from  the  Atlantic  to  the  Pacific  through  the 
northern  hemisphere.  In  the  southern  hemisphere,  the  American 
continent  extends  to  56°  of  S.  latitude,  and  in  about  latitude  60° 
commences  a  range  of  islands,  near  which  are  indications  of  an 
extensive  antarctic  continent,  leaving  a  passage  only  about  500 
miles  in  breadth.  Through  this  passage  the  motion  of  the  tidal 
wave  (as  we  shall  presently  see)  is  eastward,  and  not  westward; 
whence  we  conclude  that  the  tides  of  the  Atlantic  are  not  propa- 
gated into  the  Pacific. 

846.  Cotidal  lines. — The  phenomena  of  the  tides,  being  thus  ex- 
ceedingly complicated,  must  be  learned  chiefly  from  observations ; 
and  in  order  to  present  the  results  of  observations  most  conven- 
iently upon  a  map,  we  draw  a  line  connecting  all  those  places 
which  have  high  water  nt  the  same  instant  of  absolute  time.  Such 
lines  are  called  cotidal  lines.  The  accompanying  map,  Plate  I., 
shows  the  cotidal  lines  for  nearly  every  ocean,  drawn  at  intervals 
of  3  hours,  and  expressed  in  Greenwich  time. 


THE   TIDES. 


189 


347.  Origin  of  the  tidal  wave. — By  inspecting  tliis  map,  we  per- 
ceive that  the  great  tidal  wave  originates  in  the  Pacific  Ocean,  not 
far  from  the  western  coast  of  South  America,  in  which  region 
high  water  occurs  about  two  hours  after  the  moon  has  passed 
the  meridian.  The  wave  thus  formed,  if  left  undisturbed,  would 
travel,  like  ordinary  waves,  with  a  velocity  depending  upon  the 
depth  of  water.  When  the  breadth  of  a  wave  is  very  great  in 
comparison  with  the  depth  of  water,  the  velocity  of  its  progress 
is  equal  to  that  which  a  heavy  body  would  acquire  in  falling  by 
gravity  through  half  the  depth  of  the  liquid.  The  velocity  of 
such  a  wave  for -different  depths  of  the  ocean  is  as  follows: 


25  feet,  " 

19  miles  per  hour. 

100    " 

39     "          " 

When  the 
depth  of  the  - 
water  is 

250    " 
1,000    " 
5,000    " 

the  velocity  of  ^ 
the  wave  is 

61     "          " 
122     "          " 
273     "          " 

20,000    " 

547     "          " 

150,000    "     J 

.865     "         " 

348.  Progress  and  velocity  of  the  tidal  wave. — Since  the  moon 
travels  westward  at  the  rate  of  1000  miles  per  hour  over  the  equa 
tor,  it  tends  to  carry  high  water  along  with  it  at  the  same  rate. 
But  the  shallow  water  of  most  parts  of  the  ocean  does  not  allow 
the  tidal  wave  to  travel  with  this  velocity.  The  wave  of  high 
water,  first  raised  near  the  western  coast  of  South  America,  trav- 
els toward  the  northwest  through  the  deep  water  of  the  Pacific  at 
the  rate  of  850  miles  per  hour,  and  in  about  ten  hours  reaches  the 
coast  of  Kamtschatka.  On  account  of  more  shallow  water,  the 
same  wave  travels  westward  and  southwestward  with  less  veloc- 
ity, and  it  is  about  12  hours  old  when  it  reaches  New  Zealand, 
having  advanced  at  the  rate  of  about  400  miles  per  hour.  Pass- 
ing south  of  Australia,  the  tidal  wave  travels  westward  and  north- 
ward  into  the  Indian  Ocean,  and  is  29  hours  old  when  it  reaches 
the  Cape  of  Good  Hope.  Hence  it  is  propagated  through  the  At- 
lantic Ocean,  traveling  northward  at  the  rate  of  about  700  miles 
per  hour,  and  in  40  hours  from  its  first  formation  it  reaches  the 
shallow  waters  of  the  coast  of  the  United  States,  whence  it  is  prop- 
agated into  all  the  bays  and  inlets  of  the  coast.  The  wave  which 
enters  at  the  eastern  end  of  Long  Island  Sound  is  about  4  hours 
in  reaching  the  western  end,  so  that  the  wave  is  44  hours  old 
when  it  arrives  at  New  Haven. 


190  ASTRONOMY. 

349.  Tides  of  tiie  North  Atlantic. — A  portion  of  the  great  At- 
lantic wave  advances  up  Baffin's  Bay,  and  at  the  end  of  56  hours 
reaches  the  latitude  of  78°.     The  principal  part  of  the  Atlantic 
wave,  however,  turns  eastward  toward  the  Northern  Ocean,  and 
in  44  hours  brings  high  water  to  the  western  coast  of  Ireland. 
After  passing  Scotland,  a  portion  of  this  wave  tunfs  southward 
with  diminished  velocity  into  the  North  Sea,  and  thence  follows 
up  the  Thames,  bringing  high  water  to  London  at  the  end  of  66 
hours  from  the  first  formation  of  this  wave  in  the  Pacific  Ocean. 

350.  Velocity  of  the  tidal  wave  in  shallow  water. — As  the  tidal 
wave  approaches  the  shallow  water  of  the  coast,  its  velocity  is 
speedily  reduced  from  500,  or  perhaps  900  miles  per  hour,  to  100 
miles,  and  soon  to  30  miles  per  hour ;  and  in  ascending  bays  and 
rivers  its  velocity  becomes  still  less.    From  the  entrance  of  Chesa- 
peake Bay  to  Baltimore  the  tide  travels  at  the  average  rate  of  16 
miles  per  hour,  and  it  advances  up  Delaware  Bay  with  about  the 
same  velocity.     From  Sandy  Hook  to  New  York  city  the  tide 
advances  at  the  rate  of  20  miles  per  hour,  and  it  travels  from 
New  York  to  Albany  in  9h.  9m.,  being  at  the  average  rate  of 
nearly  16  miles  per  hour. 

From  New  York  Bay  the  tidal  wave  is  propagated  through 
East  River  until  it  meets  the  wave  which  has  come  in  from  the 
Atlantic  through  the  eastern  end  of  the  Sound.  This  place  of 
meeting  is  only  21  miles  from  New  York,  showing  that  the  veloc- 
ity of  the  tidal  wave  through  East  River  is  only  7£  miles  per 
hour — a  result  which  must  be  ascribed  to  the  narrowness  and  in- 
tricacy of  the  channel. 

351.  Tidal  wave  on  the  western  coast  of  South  America. — The 
tidal  wave  which  we  have  thus  traced  through  oceans,  bays,  and 
rivers,  has  every  variety  of  direction  ;  in  some  places  advancing 
westward,  and  in  others  eastward ;  in  some  places  northward,  and 
in  others  southward ;  but  in  each  case  it  may  be  regarded  as  a 
continuous  forward  movement,  and  the  change  in  its  direction  re- 
sults from  a  change  in  the  direction  of  the  channel.    But  there  is 
one  exception  to  this  general  rule.    We  have  traced  the  origin  of 
the  tidal  wave  to  a  region  about  1000  miles  west  of  the  coast  of 
South  America.     From  this  point  high  water  is  not  only  propa- 
gated westward  around  the  globe,  but  also  eastward  toward  Cape 


THE   TIDES.  191 

Horn.  In  this  region  the  motion  of  the  tidal  wave  appears  to  be 
similar  to  that  of  the  wave  produced  by  throwing  a  stone  upon 
the  surface  of  a  tranquil  lake,  the  wave  traveling  off  in  all  direc- 
tions from  the  first  point  of  disturbance. 

352.  Is  thftidal  wave  a  free  or  a  forced  oscillation? — If  the  moon 
should  suddenly  cease  its  disturbing  action  upon  the  waters  of  the 
ocean,  the  tidal  wave  already  formed  would  travel  forward  with 
a  velocity  depending  solely  upon  the  depth  of  water,  and  this 
would  be  called  a  free  wave.     Now  the  moon  continually  tends  to 
form  high  water  directly  beneath  it — that  is,  it  tends  to  carry  high 
water  westward  at  the  rate  of  1000  miles  per  hour  over  the  equa- 
tor.    Such  a  wave,  if  it  could  actually  be  formed,  would  be  called 
a  forced  oscillation,  because  its  velocity  would  be  independent  of 
the  depth  of  water.     Is,  then,  the  great  tidal  wave  a  free  or  a 
forced  oscillation?     We  may  answer  this  question  by  observing 
the  velocity  of  the  tidal  wave  in  the  Atlantic  Ocean,  whose  depth 
has  been  approximately  determined.     The  velocity  of  the  tidal 
wave  in  the  North  Atlantic,  from  the  equator  to  latitude  50°,  is 
about  640  miles  per  hour,  corresponding  to  a  depth  of  27,500  feet, 
which  is  somewhat  greater  than  the  average  depth  of  the  At- 
lantic.    The  velocity  of  the  tidal  wave  in  the  Atlantic  appears  to 
be  about  one  third  greater  than  that  of  a  free  wave,  and  this  ex- 
cess of  velocity  is  probably  due  to  the  immediate  action  of  the 
sun  and  moon  ;  in  other  words,  the  tidal  wave  is,  to  some  extent, 
a,  forced  oscillation,  but  its  rate  of  progress  appears  to  be  determ- 
ined mainly  by  the  depth  of  water. 

% 

353.  Height  of  the  tides. — At  small  islands  in  mid-ocean  the  tides 
never  rise  to  a  great  height,  sometimes  even  less  than  one  foot ; 
and  the  average  height  of  the  tides  for  the  islands  of  the  Atlantic 
and  Pacific  Oceans  is  only  3|  feet.     Upon  approaching  an  exten- 
sive coast  where  the  water  is  shallow,  the  velocity  of  this  tidal 
wave  is  diminished,  the  cotidal  lines  are  crowded  more  closely  to- 
gether, and  the  height  of  the  tide  is  thereby  increased;  so  that 
while  in  mid-ocean  the  average  height  of  the  tides  does  not  ex- 
ceed 3£  feet,  the  average  in  the  neighborhood  of  continents  is  not 
less  than  4  or  5  feet.     According  to  theory,  the  height  of  the 
wave  should  vary  inversely  as  the  fourth  root  of  the  depth ;  that 
is,  in  water  100  feet  deep,  the  wave  should  be  twice  as  high  as  in 


192 


ASTRONOMY. 


water  1600  feet  deep.     Fig.  91  shows  the  change  in  the  form  of 
waves  in  approaching  shallow  water. 

Fig.  91.      E  G  I  L 


A 


V    V 


K 


35-1.  Height  of  the  tides  modified  by  the  conformation  of  the  coast. — 
Along  the  coast  of  an  extensive  continent  the  height  of  the  tides 
is  greatly  modified  by  the  conformation  of  the  shore  line.  When 
the  coast  is  indented  by  broad  bays  which  are  open  in  the  direc- 
tion of  the  tidal  wave,  this  wave,  being  contracted  in  breadth, 
must  necessarily  increase  in  height,  so  that  at  the  head  of  a  bay 
the  height  of  the  tide  may  be  several  times  as  great  as  at  the  en- 
trance. The  operation  of  this  principle  is  exhibited  at  numerous 
places  upon  the  Atlantic  coast.  Thus,  if  we  draw  a  straight  line 
rig.  93.  from  Cape  Hatteras  to  the  southern 

part  of  Florida,  it  will  cut  off  a  bay 
about  200  miles  in  depth.  At  Cape 
Hatteras  and  Cape  Florida  the  tide 
rises  only  2  feet;  at  Cape  Fear  and 
St.  Augustine  it  rises  4  feet;  while 
-  at  Savannah  it  rises  7  feet 

355.  Tides  in  the  Bay  ofFundy. — If  we  draw  a  straight  line  from 
Nantucket  to  Cape  Sable,  it  will  cut  off  a  bay  in  which  the  phe- 
nomena of  the  tides  are  still  more  remarkable.  At  Nan  tucket 
the  tide  rises  only  2  feet ;  at  Boston  it  rises  10  feet ;  near  the  enr 
trance  to  the  Bay  of  Fundy,  18  feet ;  while  at  the  head  of  the  bay 
it  sometimes  rises  to  the  height  of  70  feet.  This  result  is  due 
mainly  to  the  contraction  of  the  channel  through  which  the  tidal 
wave  advances. 

Fig.  93. 


THE  TIDES. 


193 


356.  Tides  of  Long  Island  Sound,  etc. — So,  also,  at  the  east  end  of 
Long  Island  Sound,  the  tide  rises  only  2  feet ;  but  in  its  progress 
westward  through  the  Sound  the  height  continually  increases, 
until  at  the  west  end  the  height  is  more  than  7  feet 

At  the  entrance  to  Delaware  Bay  the  tide  rises  only  3£  feet, 
while  at  New  Castle  it  rises  6J  feet 

The  tide  from  the  North  Atlantic  is  propagated  through  the 
Gulf  of  St.  Lawrence,  and  thence  through  the  River  St.  Lawrence, 
at  the  average  rate  of  about  70  miles  per  hour,  being  12  hours 
from  the  ocean  to  Quebec.  This  tide  increases  in  height  as  it  ad- 
vances, being  only  9  feet  at  the  mouth  of  the  St.  Lawrence,  while 
it  is  20  feet  at  Quebec. 

357.  Tides  modified  by  a  projecting  promontory. — A  promontory, 

as  A,  projecting  into  the  ocean, 
so  as  to  divide  the  tidal  wave  and 
throw  it  off  upon  either  side,  not 
only  causes  the  tide  at  B  and  C 
to  rise  above  the  mean  height,  but 
sometimes  reduces  the  tide  at  A  below  the  mean  height.  Thus, 
at  Cape  Hatteras,  the  tide  rises  less  than  2  feet  in  height,  while 
along  the  coast  on  either  side  the  tide  rises  to  the  height  of  5  or 
6  feet.  So,  also,  on  the  south  side  of  Nantucket,  the  tides  are  less 
than  2  feet  in  height,  while  along  the  coast  north  of  Cape  Cod  the 
tide  rises  10  feet  in  height. 

358.  Tides  on  the  coast  of  Ireland. — So,  also,  on  the  southwest 

Fig.  rs.      / 


194  ASTRONOMY. 

coast  of  Ireland,  where  the  tidal  wave  from  the  Atlantic  first 
strikes  the  coast,  the  tide  is  less  than  it  is  at  a  short  distance  along 
the  coast  either  eastward  or  northward. 

In  some  cases  the  form  and  position  of  a  promontory  are  such 
as  to  divert  the  tidal  wave  from  some  part  of  the  coast,  and  leave 
it  almost  destitute  of  a  tide.  Such  a  case  occurs  on  the  east  coast 
of  Ireland.  The  wave  from  the  Atlantic,  being  forced  up  St 
George's  Channel,  is  driven  upon  the  coast  of  Wales,  where  the 
tide  rises  to  the  height  of  36  feet,  while  it  is  almost  wholly  di- 
verted from  the  opposite  coast  of  Ireland,  where  the  range  of  the 
tide  is  only  2  feet. 

359.  Tides  of  rivers. — The  tides  of  rivers  exhibit  the  operation 
of  similar  principles.    In  a  channel  of  uniform  breadth  and  depth, 
the  height  of  the  tide  should  gradually  diminish,  in  consequence 
of  the  effect  of  friction.    But  if  the  channel  contracts  or  shoals  rap- 
idly, the  height  of  the  tide  will  increase.     There  is,  then,  a  certain 
rate  of  contraction,  with  which  the  range  of  the  tides  will  remain 
stationary.     If  the  river  contracts  more  rapidly,  the  height  of  the 
tides  will  increase ;  if  the  channel  expands,  the  height  of  the  tides 
will  diminish.     Hence,  in  ascending  a  long  river,  it  may  happen 
that  the  height  of  the  tides  will  increase  and  decrease  alternately. 

Thus,  at  New  York,  the  mean  height  of  the  tide  is  4.3  feet ;  at 
"West  Point,  55  miles  up  the  Hudson  River,  the  tide  rises  only  2.7 
feet ;  at  Tivoli,  98  miles  from  New  York,  the  tide  amounts  to  4 
feet;  while  at  Albany  it  rises  only  2.3  feet 

360.  The  diurnal  inequality  in  the  height  of  the  tides. — If  the  sun 
and  moon  moved  always  in  the  plane  of  the  equator,  and  the 
earth  were  entirely  covered  with  water  to  a  great  depth,  the  two 
daily  tides  should  have  nearly  the  same  height ;  but  when  they 
are  out  of  the  equator,  the  two  daily  tides  should  generally  be 
unequal.     The  moon  sometimes  reaches  28°  north  declination,  in 
which  case  it  tends  to  raise  the  highest  tide  at  a  station  in  latitude 
28°  north,  while  the  highest  tide  on  the  opposite  side  of  the  earth 
should  be  in  latitude  28°  south.     Hence  the  two  tides  which  are 
formed  in  the  northern  hemisphere  under  opposite  meridians 
must  be  of  unequal  heights — that  is,  the  morning  and  evening 
tides  at  a  given  place  should  be  unequal.     The  same  would  be 
true  for  the  southern  hemisphere,  but  on  the  equator  there  would 
be  no  such  diurnal  inequality. 


THE  TIDES. 


195 


361.  Diurnal  inequality  in  the  North  Atlantic  Ocean. — Along  the 
Atlantic  coast  of  the  United  States,  when  the  moon  has  its  great- 
est declination,  the  difference  between  high  water  in  the  forenoon 
and  afternoon  averages  about  18  inches  j  but  at  Boston  this  dif- 
ference sometimes  amounts  to  36  inches. 

On  the  coast  of  Ireland,  the  diurnal  inequality,  at  its  maximum, 
is  only  one  foot,  while  the  average  height  of  the  tides  is  nine  feet. 
On  some  parts  of  the  European  coast  the  diurnal  inequality  is 
still  smaller,  and  can  with  difficulty  be  detected  in  a  long  series 
of  observations. 

362.  Diurnal  inequality  on  the  Pacific  coast. — On  the  Pacific  coast 
of  the  United  States,  when  the  moon  is  far  from  the  equator,  there 
is  one  large  and  one  small  Fijr.ro. 

tide  during  each  day.  In 
the  Bay  of  San  Francisco, 
the  difference  between  high  7  I/T\ 

and  low  water  in  the  fore- 
noon is  sometimes  only  two  s  U,  /  \  / 
inches,  while  in  the  afternoon 
of  the  same  day  the  differ- 
ence is  5£  feet.  When  the 
moon  is  on  the  equator  this 

inequality    disappears,   and  ^       / 

the  two  daily  tides  are  near- 
ly equal. 

At  other  places  on  the  Pacific  coast  this  inequality  in  the  two 
daily  tides  is  still  more  re- 
markable.  At  Port  Town- 
send,  near  Vancouver's  Isl- 
and, when  the  moon  has  its 
greatest  declination,  there  is 
no  descent  corresponding  to 
morning  low  water,  but  mere- 
ly a  temporary  check  in  the 
rise  of  the  tide.  Thus  one  of 
the  two  daily  tides  becomes 
obliterated ;  that  is,  we  find 
but  one  tide  in  the  24  hours. 
Similar  phenomena  occur  at 


HO  UR  S  . 
t   A   e  a  10  iz  tit  »B  is  so  iz  **•  z 

F&T 

1 

/ 

"^ 

e 

x 

x 

s 

X 

^~~ 

\ 

/ 

it 

2 

2. 

\ 

/ 

1 

V 

; 

0 

^ 

196  ASTRONOMY. 

other  places  upon  the  Pacific  coast,  and  also  on  tb»  coast  of 
Kamtschatka. 

363.  Cause  of  these  variations  in  the  diurnal  inequality. — The  tide 
actually  observed  at  any  port  is  the  effect,  not  simply  of  the  im- 
mediate action  of  the  sun  and  moon  upon  the  waters  of  the  ocean, 
but  is  rather  the  resultant  of  their  continued  action  upon  the  wa- 
ters of  the  different  seas  through  which  the  wave  has  advanced 
from  its  first  origin  in  the  Pacific  until  it  reaches  the  given  port, 
embracing  an  interval  sometimes  of  one  or  two  days,  and  perhaps 
even  longer.     During  this  period  the  moon's  action  tends  some- 
times to  produce  a  large  tide,  and  sometimes  a  small  one ;  and  in 
a  tide  whose  age  is  more  than  12  hours,  these  different  effects  are 
combined  so  as  sometimes  partly  to  obliterate  the  diurnal  ine- 
quality, and  sometimes  to  exaggerate  it.     This  is  probably  the 
reason  why  the  diurnal  inequality  is  less  noticeable  in  the  North 
Atlantic  than  in  the  North  Pacific. 

364.  Four  tides  in  24  hours. — In  some  places  the  tide  rises  and 
falls  four  times  in  2-i  hours.     This  happens  on  the  east  coast  of 
Scotland,  where  the  form  of  the  tidal  wave  is  such  as  is  repre- 

Fig.98.  sentedby  the  annexed  fig- 

ure. This  anomaly  is  as- 
cribed to  the  superposition 
of  two  tidal  waves,  one 
traveling  round  the  north 
of  Scotland,  and  advanc- 
ing southward  through  the  North  Sea,  while  the  other  passes 
through  the  English  Channel,  and  thence  advances  northward 
into  the  same  sea.  At  some  places  these  two  waves  arrive  near- 
ly at  the  same  hour,  and  are  so  superposed  as  not  to  be  distin- 
guished from  each  other ;  but  at  other  places  one  arrives  2  or  3 
hours  behind  the  other,  thus  presenting  the  appearance  of  high 
water  4  times  in  24  hours. 

365.  Small  tides  of  the  Pacific  Ocean.— Near  the  middle  of  the 
Pacific  Ocean,  in  the  neighborhood  of  the  Society  Islands,  from 
latitude  13°  to  18°  S.,  and  from  longitude  140°  to  176°  W.,  the 
tides  are  smaller  than  have  been  found  in  any  other  portion  of 
the  open  sea,  averaging  less  than  one  foot  in  height     At  Tahiti 


THE   TIDES.  197 

(latitude  17°  29'  S.,  longitude  149°  29'  W.),  the  tides  at  full  moon 
rise  to  the  height  of  about  15  inches,  and  at  the  quadratures  only 
about  3  inches.  There  are  two  high  waters  daily  occurring  near 
noon  and  midnight,  being  seldom  earlier  than  10  A.M.,  or  later 
than  2J  PM. 

366.  Cause  of  these  peculiarities.  —  It  is  uncertain  what  is  the 
cause  of  this  small  height  of  the  tides,  but  it  is  believed  that  the 
following  consideration  will  explain  it,  at  least  in  part.     The 
original  tide  wave,  starting  from  the  eastern  part  of  the  Pacific 
Ocean,  reaches  Tahiti  about  six  hours  after  the  moon's  transit 
over  that  meridian.     Hence,  when  the  main  tidal  wave  of  the 
Pacific  reaches  that  port,  the  immediate  effect  of  the  moon  is  to 
produce  low  water  at  the  same  hour;  and  the  superposition  of 
these  two  waves  produces  a  nearly  uniform  level  of*the  water. 

The  occurrence  of  high  water  within  about  two  hours  of  noon 
every  day  seems  to  indicate  that  the  power  of  the  sun  to  raise  a 
tide  is  here  nearly  equal  to  that  of  the  moon.  In  the  Atlantic 
Ocean,  the  influence  of  the  moon  upon  the  tides  is  generally  about 
double  that  of  the  sun ;  but  this  ratio  appears  to  be  a  variable 
one. 

367.  Tides  of  the  G u2f  of  Mexico.— The  Gulf  of  Mexico  is  a  shal- 
low sea,  about  800  miles  in  diameter,  almost  entirely  surrounded 
by  land,  and  communicating  with  the  Atlantic  by  two  channels, 
each  about  100  miles  in  breadth.     It  is  by  the  Florida  channel 
that  the  tidal  wave  from  the  Atlantic  is  chiefly  propagated  into 
the  Gulf,  but  its  progress  is  so  much  obstructed  by  the  West  In- 
dia Islands  that  its  height  is  very  much  reduced.    Between  Flor- 
ida and  Cuba  the  tidal  wave  advances  slowly  westward ;  but  after 
passing  the  channel  it  moves  more  rapidly,  and  reaches  the  west- 
ern side  of  the  Gulf  in  seven  hours,  showing  an  average  progress 
of  125  miles  per  hour. 

The  tides  in  the  Gulf  are  every  where  quite  small.  At  Mobile 
and  Pensacola  the  average  height  is  only  one  foot  The  diurnal 
inequality  is  also  quite  large,  so  that  at  most  places  (except  when 
the  moon  is  near  the  equator)  one  of  the  daily  tides  is  well-nigh 
inappreciable,  and  the  tide  is  said  to  ebb  and  flow  but  once  in  24 
hours. 


198  ASTRONOMY. 

368.  Tides  of  the  Mediterranean. — The  tides  of  the  Mediterranean 
are  generally  so  small  as  not  to  be  regarded  by  navigators.    Their 
average  height  does  not  exceed  18  inches.    In  the  neighborhood 
of  the  Strait  of  Gibraltar  the  tide  rises  from  2  to  4  feet ;  at  Venice 
it  rises  from  18  inches  to  4  feet;  and  at  Tunis  it  sometimes  rises 
to  the  height  of  3  feet. 

The  length  of  the  Mediterranean  is  2400  miles,  or  nearly  one 
third  the  diameter  of  the  earth ;  and  the  average  height  of  the 
tides  is  here  at  least  one  third  what  it  is  in  the  open  sea. 

369.  Tides  of  inland  seas. — In  small  lakes  and  seas  which  do  not 
communicate  with  the  ocean  there  is  a  daily  tide,  but  so  small 
that  it  requires  the  most  accurate  observations  to  detect  it.     The 
existence  of  a  tide  in  Lake  Michigan  has  been  proved  by  a  series 
of  observations  made  at  Chicago  in  1859.     The  average  height 
of  this  tide  is  If  inches ;  and  the  average  time  of  high  water  is  30 
minutes  after  the  time  of  the  moon's  transit. 

The  length  of  Lake  Michigan  is  350  miles,  or  -^d  of  the  earth's 
diameter,  and  its  tide  is  about  -^jd  of  that  which  prevails  in  mid- 
ocean. 


CHAPTER  XIY. 

THE   PLANETS  —  THEIR   APPARENT   MOTIONS.  — ELEMENTS    OF 
THEIR  ORBITS. 

370.  Number,  etc.,  of  the  planets. — The  planets  are  bodies  of  a 
globular  form,  which  revolve  around  the  sun  as  a  common  cen- 
tre, in  orbits  which  do  not  differ  much  from  circles.  The  name 
planet  is  derived  from  the  Greek  word  TrAavjjrijc,  signifying  a 
wanderer,  and  was  applied  by  the  ancients  to  these  bodies  be- 
cause their  apparent  movements  were  complicated  and  irregular. 
Five  of  the  planets— Mercury,  Venus,  Mars,  Jupiter,  and  Saturn- 
are  very  conspicuous,  and  have  been  known  from  time  immemo- 
rial. Uranus  was  discovered  in  1781,  and  Neptune  in  1846,  mak- 
ing eight  planets  including  the  earth.  Besides  these  there  is  a 
large  group  of  small  planets,  called  asteroids,  situated  between  the 
orbits  of  Mars  and  Jupiter.  The  first  of  these  was  discovered  in 
1801,  and  the  number  known  in  1877  amounted  to  178. 

The  orbits  of  Mercury  and  Venus  are  included  within  the  orbit 


THE  PLANETS,  ETC.  199 

of  the  earth,  and  they  are  hence  called  inferior  planets,  while  the 
others  are  called  superior  planets. 

371.  The  satellites. — Some  of  the  planets  are  the  centres  of  sec- 
ondary systems,  consisting  of  smaller  globes,  revolving  round 
them  in  the  same  manner  as  they  revolve  around  the  sun.     These 
are  called  satellites  or  moons.    The  primary  planets  which  are  thus 
attended  by  satellites  carry  the  satellites  with  them  in  their  or- 
bits around  the  sun.     Of  the  satellites  known  at  the  present  time, 
two  revolve  around  Mars,  four   around  Jupiter,  eight   around 
Saturn,  four  around  Uranus,  and  one  around  Neptune.     The 
moon  is  also  a  satellite  to  the  earth. 

372.  The  orbits  of  the  planets.— The  orbit  of  each  of  the  planets 
is  an  ellipse,  of  which  the  sun  occupies  one  of  the  foci.     That 
point  of  its  orbit  at  which  a  planet  is  nearest  the  sun  is  called 
the  perihelion,  and  that  point  at  which  it  is  most  remote  is  called 
the  aphelion. 

The  eccentricity  of  a  planetary  orbit  is  the  distance  of  the  sun 
from  the  centre  of  the  ellipse  which  the  planet  describes,  expressed 
in  terms  of  the  semi-major  axis  regarded  as  a  unit;  or,  in  other 
words,  it  is  the  quotient  of  the  distance  between  the  centre  and 
focus,  divided  by  the  semi-major  axis.  The  eccentricities  of  most 
of  the  planetary  orbits  are  so  minute  that,  if  the  form  of  the  orbit 
were  exactly  delineated  on  paper,  it  could  not  be  distinguished 
from  a  circle  except  by  careful  measurement. 

373.  Geocentric  and  heliocentric  places. — The  motion  of  a  planet 
as  it  appears  to  an  observer  on  the  earth  is  called  the  geocentric 
motion,  while  its  motion  as  it  would  appear  if  the  observer  were 
transferred  to  the  sun  is  called  its  heliocentric  motion.     The  mo- 
tions of  the  planets  can  not  be  observed  from  the  sun  as  a  centre, 
but  from  the  geocentric  motions,  combined  with  the  relative  dis- 
tances of  the  earth  and  planet  from  the  sun,  we  may  deduce  the 
heliocentric  motions  by  the  principles  of  Geometry. 

The  geocentric  place  of  a  body  is  its  place  as  seen  from  the  cen- 
tre of  the  earth,  and  the  heliocentric  place  is  its  place  as  seen  from 
the  centre  of  the  sun. 

374.  Elongation,  conjunction,  and  opposition  of  a  planet. — The 


200  -        ASTRONOMY. 

angle  formed  by  lines  drawn  from  the  earth  to  the  sun  and  a 
planet  is  called  the  elongation  of  the  planet  from  the  sun ;  and  it 
is  east  or  west,  according  as  the  planet  is  on  the  east  or  west  side 
of  the  sun. 

A  planet  is  said  to  be  in  conjunction  with  the  sun  when  it  has 
the  same  longitude,  being  then  in  nearly  the  same  part  of  the 
heavens  with  the  sun.  It  is  said  to  be  in  opposition  with  the  sun 
when  its  longitude  differs  from  that  of  the  sun  180°,  being  then 
in  the  quarter  of  the  heavens  opposite  to  the  sun.  A  planet  is 
said  to  be  in  quadrature  when  it  is  distant  from  the  sun  90°  in 
longitude. 

A  planet  which  is  in  conjunction  with  the  sun  passes  the  me- 
ridian about  noon,  and  is  therefore  above  the  horizon  only  dur- 
ing the  day.  A  planet  which  is  in  opposition  with  the  sun  passes 
the  meridian  about  midnight,  and  is  therefore  above  the  horizon 
during  the  night  A  planet  which  is  in  quadrature  passes  the 
meridian  about  6  o'clock  either  morning  or  evening. 

An  inferior  planet  is  in  conjunction  with  the  sun  when  it  is  be- 
tween the  earth  and  the  sun,  as  well  as  when  it  is  on  the  side  of 
the  sun  opposite  to  the  earth.  The  former  is  called  the  inferior 
conjunction,  the  latter  the  superior  conjunction. 

375.  Why  the  apparent  motions  of  the  planets  differ  from  the  real 
motions. — If  the  planets  could  be  viewed  from  the  sun  as  a  centre, 
they  would  all  be  seen  to  advance  invariably  in  the  same  direc- 
tion, viz.,  from  west  to  east,  in  planes  only  slightly  inclined  to  each 
other,  but  with  very  unequal  velocities.  Mercury  would  advance 
eastward  with  a  velocity  about  one  third  as  great  as  our  moon ; 
Venus  would  advance  in  the  same  direction  with  a  velocity  less 
than  half  that  of  Mercury;  the  more  distant  planets  would  ad- 
vance still  more  slowly ;  while  the  motions  of  Uranus  and  Nep- 
tune would  be  scarcely  appreciable  except  by  comparing  observa- 
tions made  at  long  intervals  of  time.  None  of  the  planets  would 
ever  appear  to  move  from  east  to  west. 

The  motions  of  the  planets,  as  they  actually  appear  to  us,  are 
very  unlike  those  just  described,  first,  because  we  view  them  from 
a  point  remote  from  the  centre  of  their  orbits,  in  consequence  of 
which  the  distances  of  the  planets  from  the  earth  are  subject  to 
great  variations;  and,  second,  because  the  earth  itself  is  in  motion, 
and  the  planets  have  an  apparent  motion,  resulting  from  the  reaJ 
motion  of  the  earth. 


APPARENT  MOTIONS  OF  THE  PLANETS. 


201 


376.  The  apparent  motion  of  an  inferior  planet. — In  order  to  de- 
duce the  apparent  motion  of  an  inferior  planet  from  its  real  mo- 
tion, let  CKZ  represent  a  portion  of  the  heavens  lying  in  the  plane 
of  the  ecliptic ;  let  a,  b,  c,  d,  etc.,  be  the  orbit  of  the  earth  ;  and 


AE 


1,  2,  3, 4,  etc.,  the  orbit  of  Mercury.  Let  the  orbit  of  Mercury  be 
divided  into  12  equal  parts,  each  of  which  is  described  in  7^- 
days ;  and  let  «5,  be,  cd,  etc.,  be  the  spaces  described  by  the  earth, 
in  the  same  time.  Suppose  Mercury  to  be  at  the  point  1  in  his 
orbit  when  the  earth  is  at  the  point  a;  Mercury  will  then  appear 
in  the  heavens  at  A,  in  the  direction  of  the  line  al.  In  7^-  days 
Mercury  will  have  arrived  at  2,  while  the  earth  has  arrived  at  b, 
and  therefore  Mercury  will  appear  at  B.  When  the  earth  is  at  c, 
Mercury  will  appear  at  C,  and  so  on.  By  laying  the  edge  of  a 
ruler  on  the  points  c  and  3,  d  and  4,  e  and  5,  and  so  on,  the  suc- 
cessive apparent  places  of  Mercury  in  the  heavens  will  be  ob- 
tained. We  thus  find  that  from  A  to  C,  his  apparent  motion  is 


202  ASTRONOMY. 

from  east  to  west ;  from  C  to  P,  his  apparent  motion  is  from  west 
to  east ;  from  P  to  T  it  is  from  east  to  west ;  and  from  T  to  Z  the 
apparent  motion  is  from  west  to  east. 

377.  Direct  and  retrograde  motion. — When  a  planet  appears  to 
move  in  the  direction  in  which  the  sun  appears  to  move  in  the 
ecliptic,  its  apparent  motion  is  said  to  be  direct;,  and  when  it  ap- 
pears to  move  in  the  contrary  direction,  it  is  said  to  be  retrograde. 
The  apparent  motion  of  an  inferior  planet  is  always  direct,  except 
•within  a  certain  elongation  east  and  west  of  the  inferior  conjunc- 
tion, when  it  is  retrograde. 

If  we  follow  the  movements  of  Mercury  during  several  success- 
ive revolutions,  we  shall  find  its  apparent  motion  to  be  such  as  is 
indicated  by  the  arrows  in  the  preceding  diagram,  viz.,  while  pass- 
ing from  its  greatest  western  to  its  greatest  eastern  elongation,  it 
appears  to  move  in  the  same  direction  as  the  sun  toward  P.  As 
it  approaches  P  its  apparent  motion  eastward  becomes  gradually 
slower,  until  it  stops  altogether  at  P,  and  becomes  stationary.  It 
then  moves  westward,  returning  to  T,  where  it  again  becomes  sta- 
tionary, after  which  it  again  moves  eastward,  and  continues  to 
move  in  that  direction  through  an  arc  about  equal  to  CP,  when 
it  again  becomes  stationary.  It  again  moves  westward  through 
an  arc  about  equal  to  PT,  when  it  again  becomes  stationary,  and 
so  on.  The  middle  point  of  the  arc  of  retrogression,  PT,  is  that 
at  which  the  planet  is  in  inferior  conjunction ;  and  the  middle 
point  of  the  arc  of  progression,  CP,  is  that  at  which  the  planet  is 
in  superior  conjunction. 

These  apparently  irregular  movements  suggested  to  the  an- 
cients the  name  of  planet,  or  wanderer. 

378.  Apparent  motion  of  a  superior  planet. — In  order  to  deduce 
the  apparent  motion  of  a  superior  planet  from  the  real  motions  of 
the  earth  and  planet,  let  S  be  the  place  of  the  sun ;  1,  2, 3,  etc.,  be 
the  orbit  of  the  earth ;  a,  i,  c,  etc.,  the  orbit  of  Mars ;  and  CGL  a 
part  of  the  starry  firmament.     Let  the  orbit  of  the  earth  be  di- 
vided into  12  equal  parts,  each  of  which  is  described  in  one  month ; 
and  let  ab,  be,  cd,  etc.,  be  the  spaces  described  by  Mars  in  the  same 
time.    Suppose  the  earth  to  be  at  the  point  1  when  Mars  is  at  the 
point  a,  Mars  will  then  appear  in  the  heavens  in  the  direction  of 
the  line  1  a.     When  the  earth  is  at  3  and  Mars  at  c,  he  will  ap- 


APPARENT  MOTIONS  OF  THE  PLANETS.  203 

Fig.  100. 


pear  in  the  heavens  at  C.  When  the  earth  arrives  at  4,  Mars  will 
arrive  at  d,  and  will  appear  in  the  heavens  at  D.  While  the  earth 
moves  from  4  to  5  and  from  5  to  6,  Mars  will  appear  to  have  ad- 
vanced among  the  stars  from  D  to  E  and  from  E  to  F,  in  the  di- 
rection from  west  to  east.  During  the  motion  of  the  earth  from 
6  to  7  and  from  7  to  8,  Mars  will  appear  to  go  backward  from  F 
to  G  and  from  G  to  H,  in  the  direction  from  east  to  west.  Dur- 
ing the  motion  of  the  earth  from  8  to  9  and  from  9  to  10,  Mars 
will  appear  to  advance  from  H  to  I  and  from  I  to  K,  in  the  di- 
rection from  west  to  east,  and  the  motion  will  continue  in  the 
same  direction  until  near  the  succeeding  opposition. 

The  apparent  motion  of  a  superior  planet  projected  on  the  heav- 
ens is  thus  seen  to  be  similar  to  that  of  an  inferior  planet,  except 
that,  in  the  latter  case,  the  retrogression  takes  place  near  inferior 
conjunction,  and  in  the  former  it  takes  place  near  opposition. 


204  ASTRONOMY. 

379.  Conditions  under  which  a  planet  is  visible. — One  or  two  of 
the  planets  are  sometimes  seen  when  the  sun  is  above  the  hori- 
zon ;  but  generally,  in  order  to  be  visible  without  a  telescope,  a 
planet  must  have  an  elongation  from  the  sun  greater  than  30°,  so 
as  to  be  above  the  horizon  before  the  commencement  of  the  morn- 
ing twilight,  or  after  the  close  of  the  evening  twilight. 

The  greatest  elongation  of  the  inferior  planets  never  exceeds 
47°.  If  they  have  eastern  elongation,  they  pass  the  meridian  in 
the  afternoon,  and,  being  visible  above  the  horizon  after  sunset, 
are  called  evening  stars.  If  they  have  western  elongation,  they 
pass  the  meridian  in  the  forenoon,  and,  being  visible  above  the 
eastern  horizon  before  sunrise,  are  called  morning  stars. 

A  superior  planet,  having  every  degree  of  elongation  from  0  to 
180°,  may  pass  the  meridian  at  any  hour  of  the  day  or  night  At 
opposition  the  planet  passes  the  meridian  at  midnight,  and  is 
therefore  visible  from  sunset  to  sunrise. 

d380.  Phases  of  a  planet. — That  hemisphere  of  a  planet  which  is 
presented  to  the  sun  is  illumined,  and  the  other  is  dark.  But  if 
the  same  hemisphere  which  is  turned  toward  the  sun  is  not  also 
presented  to  the  earth,  the  hemisphere  of  the  planet  which  is  pre- 
sented to  the  earth  will  not  be  wholly  illumined,  and  the  planet 
will  exhibit  phases. 

The  inferior  planets  exhibit  the  same  variety  of  phases  as  the 
moon.  At  the  inferior  conjunction,  the  dark  side  of  the  planet  is 
turned  directly  toward  the  earth.  Soon  afterward  the  planet  ap- 
pears a  thin  crescent,  which  increases  in  breadth  until  the  great- 
est elongation,  when  it  becomes  a  half  moon.  Subsequently  its 
form  becomes  gibbous,  and  at  superior  conjunction  it  becomes  a 
full  moon. 

The  distances  of  the  superior  planets  from  the  sun  are,  with  but 
one  exception,  so  much  greater  than  that  of  the  earth,  that  the 
hemisphere  which  is  turned  toward  the  earth  is  sensibly  the  same 
as  that  turned  toward  the  sun,  and  these  planets  always  appear  full. 

381.  Elements  of  the  orlit  of  a  planet. — There  are  seven  different 
quantities  necessary  to  be  known  in  order  to  compute  the  place 
of  a  planet  for  a  given  time.  These  are  called  the  Elements  of  the 
orbit.  They  are, 

1.  The  periodic  time. 


ORBITS   OF  THE   PLANETS.  205 

2.  The  mean  distance  from  the  sun,  or  the  semi-major  axis  of 
the  orbit 

3.  The  longitude  of  the  ascending  node. 

4.  The  inclination  of  the  plane  of  the  orbit  to  that  of  the  eclip- 
tic. 

5.  The  eccentricity  of  the  orbit. 

6.  The  longitude  of  the  perihelion. 

7.  The  place  of  the  planet  in  its  orbit  at  a  particular  epoch. 

If  the  mass  of  a  planet  is  either  known  or  neglected,  the  mean 
distance  can  be  computed  from  the  periodic  time  by  means  of 
Kepler's  third  law,  so  that  the  number  of  independent  elements 
is  reduced  to  six. 

The  orbits  of  the  planets  can  not  be  determined  in  the  same 
manner  as  the  orbit  of  the  moon,  Art.  207,  because  the  centre  of 
the  earth  may  be  regarded  as  a  fixed  point  relative  to  the  moon's 
orbit,  but  it  is  not  fixed  relative  to  the  planetary  orbits.  The 
methods  therefore  employed  for  determining  the  orbits  of  the 
planets  are  in  many  respects  quite  different  from  those  which  are 
applicable  to  determining  the  orbit  of  the  moon,  and  also  that  of 
the  earth. 

382.  To  find  the  periodic  time.  First  method. — Each  of  the  plan- 
ets, during  about  half  its  revolution  around  the  sun,  is  found  to 
be  on  one  side  of  the  ecliptic,  and  during  the  other  half  on  the 
other  side.  The  period  which  elapses  from  the  time  that  a  planet 
is  at  one  of  its  nodes,  till  it  returns  to  the  same  node  (allowance 
being  made  for  the  motion  of  the  nodes),  is  the  sidereal  period  of 
the  planet.  When  a  planet  is  at  either  of  its  nodes,  it  is  in  the 
plane  of  the  ecliptic,  and  its  latitude  is  then  nothing.  Let  the 
right  ascension  and  declination  of  a  planet  be  observed  on  several 
successive  days,  near  the  period  when  it  is  passing  a  node,  and  let 
its  corresponding  longitudes  and  latitudes  be  computed.  From 
these  we  may  obtain,  by  a  proportion,  the  time  when  the  planet's 
latitude  is  nothing.  If  similar  observations  are  made  when  the 
planet  passes  the  same  node  again,  we  shall  have  the  time  of  a 
revolution. 

Example. — The  planet  Mars  was  observed  to  pass  its  ascending 
node  as  follows : 

1862,  December,    5d.  22h.  17m. 
1864,  October,     22d.  21h.  58m.   ' 


206  ASTRONOMY. 

The  interval  is  686.986  days,  which  differs  but  a  few  minutes 
from  the  most  accurate  determination  of  its  period. 

When  the  orbit  of  a  planet  is  but  slightly  inclined  to  the  eclip- 
tic, a  small  error  in  the  observations  has  a  great  influence  on  the 
computed  time  of  crossing  the  ecliptic.  A  more  accurate  result 
will  be  obtained  by  employing  observations  separated  by  a  long 
interval,  and  dividing  this  interval  by  the  number  of  revolutions 
of  the  planet 

383.  Second  method. — The  synodical  period  of  a  planet  is  the  in- 
terval between  two  consecutive  oppositions,  or  two  conjunctions 
of  the  same  kind.  The  sidereal  period  may  be  deduced  from 
the  synodical  by  a  method  similar  to  that  of  Art  205.  Let  p  be 
the  sidereal  period  of  a  planet,  p'  the  sidereal  period  of  the  earth, 
and  s  the  time  of  a  synodic  revolution,  all  expressed  in  mean  solar 
days.  The  daily  motion  of  the  planet,  as  seen  from  the  sun,  is 

— ,  while  that  of  the  earth  is  — — ;  and  if  p  be  a  superior  planet, 

360°    360° 
the  earth  will  gain  upon  the  planet  daily  — -f .     But  in  a 

synodic  revolution  the  earth  gains  upon  the  planet  360° ;  that  is, 
its  daily  gain  is .    Hence  we  have  the  equation 

360    360_360 
p'       p  ~   s  ' 
Hence  sp—s2)'=pp'1 

or  p= — — -.. 

s-p' 

For  an  inferior  planet,  we  shall  find  in  like  manner 
p~7+fi' 

384  How  to  obtain  the  mean  synodic  period. — Since  the  angular 
motion  of  the  planets  is  not  uniform,  the  interval  between  two 
successive  oppositions  will  not  generally  give  the  mean  synodical 
period.  But  if  we  take  two  oppositions,  separated  by  a  long  in- 
terval, when  the  planet  was  found  in  the  same  position  relatively 
to  some  fixed  star,  and  divide  the  interval  by  the  number  of  rev- 
olutions, we  may  obtain  the  mean  synodical  period  very  accu- 
rately. 


ORBITS  OF  THE   PLANETS. 


207 


Example. — The  planet  Mars  was  observed  in  opposition  as  fol- 
lows: 

1864,  November,  30d.  17h.  58m. 
1817,  December,     8d.   9h.  15m. 

The  interval  is  17159.37  days,  which  divided  by  22,  the  num- 
ber of  synodic  revolutions,  gives  for  the  mean  time  of  one  sy- 
nodic revolution  779.97  days.  By  comparing  the  observations 
of  Ptolemy,  A.D.  130,  with  recent  observations,  the  time  of  one 
synodical  revolution  is  found  to  be  779.936  days;  from  which, 
according  to  the  formula  given  above,  the  mean  sidereal  period 
of  Mars  is  found  to  be  686.980  days.  And  in  the  same  manner 
the  periods  of  the  other  planets  may  be  found. 

The  following  table  shows  the  time  of  a  synodical,  as  well  as 
of  a  sidereal  revolution  of  the  planets : 


Synodical 
Revolution. 

Sidereal  Revolution. 

Mean  daily  Motion. 

Mercury  .  . 
Venus   .  .  . 
Earth.  .  .  . 
Mars  .... 
Jupiter  .  .  . 
Saturn  .  .  . 
Uranus  .  .  . 
Neptune  .  . 

Days. 
115.877 

583.921 

779.936 
398.884 
378.092 
369.656 
367.489 

Days. 

87.969  or  3  months. 
224.701      7i      " 
365.256          1  year. 
686.980          2  years. 
4332.585        12     " 
10759.220        29     " 
30686.821        84    " 
60126.722       164     " 

4°    5'  32".6 
1    36     7  .8 
0    59     8  .3 
0    31    26  .7 
0      4   59  .3 
0      2     0  .6 
0      0   42  .4 
0      0   21  .6 

385.  To  find  the  distance  of  a  planet  from  the  sun. — The  mean 
distance  of  a  planet,  whose  periodic  time  is  known,  can  be  com- 
puted by  Kepler's  third  law.  It  can,  however,  be  determined  in- 
dependently by  methods  like  the  following : 

The  distance  of  an  inferior  planet  from  the  sun  may  be  determ- 
ined by  observing  the  angle  of  greatest  elongation. 

In  the  triangle  SEV,  let  S  be  the  place  of  the  sun,         ng.  101. 
E  the  earth,  and  V  an  inferior  planet  at  the  time  of 
its  greatest  elongation.     Then,  since  the  angle  SYE 
is  a  right  angle,  we  have 

SV  :  SE  ::  sin.  SEY:  radius; 
or  SY  =  SEsin.SEY. 

If  the  orbits  of  the  planets  were  exact  circles,  this 
method  would  give  the  mean  distance  of  the  planet 
from  the  sun ;  but  since  this  is  not  the  case,  we  must 
observe  the  greatest  elongation  in  different  parts  of 


208  ASTBONOMY. 

the  orbit,  and  thus  obtain  its  average  value.  The  average  value 
of  the  greatest  elongation  of  Venus  is  46°  20';  whence  the  mean 
distance  of  Venus  is  found  to  be  .7233,  the  distance  of  the  earth 
from  the  sun  being  called  unity. 

886.  Distance  of  a  superior  planet. — The  distance  of  a  superior 
planet,  whose  periodic  time  is  known,  may  be  found  by  measur- 
Fig.  102.  ing  the  retrograde  motion  of  the 

planet  in  one  day  at  the  time  of 
opposition.  Let  S  be  the  place 
of  the  sun,  E  the  earth,  and  M  the  planet  on  the  day  of  opposi- 
tion, when  the  three  bodies  are  situated  in  the  same  straight  line. 
Let  EE'  represent  the  earth's  motion  in  one  day  from  opposition, 
and  MM'  that  of  the  planet  in  the  same  time.  The  angles  ESE' 
and  MSM'  are  known  from  the  periodic  times.  Draw  E'B  par- 
allel to  SM ;  join  E'M',  and  produce  the  line  to  meet  SM  in  A. 
The  angle  SAE',  which  equals  AE'B,  is  the  retrogradation  of  the 
planet  in  one  day,  and  is  supposed  to  be  known  from  observa- 
tions. In  the  triangle  E'SM',  the  side  E'S  and  the  angle  E'SM' 
are  known,  and  E'M'S=M'SA+M'AS;  from  these  we  can  com- 
pute SM'. 

If  we  only  know  the  periodic  time  of  the  planet,  we  are  obliged, 
in  the  first  approximation,  to  assume  the  orbit  to  be  a  circle  in 
order  to  compute  the  angle  MSM' ;  but  if  we  observe  the  retro- 
grade motion  at  a  large  number  of  oppositions  in  different  parts 
of  the  orbit,  we  may  obtain  the  average  value  of  the  arc  of  retro- 
gradation,  and  hence  we  may  compute  the  mean  distance. 

Example.  The  average  arc  of  retrogradation  of  Mars  on  the  day 
of  opposition  is  21'  25".7.  If  we  take  the  mean  daily  motions  of 
the  earth  and  Mars,  as  given  on  page  207,  we  shall  find  the  mean 

distance  of  Mars  to  be  *™'^',^'  =1.52369,  the  distance  of  the 
sin.  o2  52 "A 

earth  from  the  sun  being  called  unity. 

The  following  table  shows  the  mean  distances  of  the  planets 
from  the  sun,  expressed  in  miles,  and  also  their  relative  distances, 
the  distance  of  the  earth  being  called  unity : 


ORBITS   OF   THE    PLANETS. 


209 


Msan  Distance  from  the  Sun. 

Relative  Distance. 

Mercury  .  . 

37,000,000  miles. 

0.387 

Venus   .  .  . 

69,000,000 

0.723 

Earth.  .  .  . 

95,000,000 

1.000 

Mars  .... 

145,000,000 

1.524 

Jupiter  .  .  . 

496,000,000 

5.203 

Saturn  .  .  . 

909,000,000 

9.539 

Uranus.  .  . 

1,828,000,000 

19.183 

Neptune  .  . 

2,862,000,000 

30.037 

387.  Diameters  of  the  planets. — Having  determined  the  distances 
of  the  planets,  it  is  only  necessary  to  measure  their  apparent  di- 
ameters, and  we  can  easily  compute  their  absolute  diameters  in 
miles.  The  apparent  diameters  of  the  planets  are  of  course  vari- 
able, since  they  depend  upon  the  distances  which  are  continually 
varying.  The  following  table  shows  the  mean  apparent  diame- 
ters, and  also  the  absolute  diameters  of  the  planets,,  as  well  as 
their  volumes,  that  of  the  earth  being  called  unity : 


Equatorial  Diameters. 

Volume. 

Apparent 

In  Miles. 

Mercury  .  .  . 
Venus  .... 
Earth  

7" 
17 

7 
38 
17 
4 
2 

3,000 
7,700 
7,926 
4,500 
92,000 
75,000 
36,000 
35,000 

TT 
9 
To" 

\ 

1412 
770 
96 
90 

Mars  

Jupiter  .... 
Saturn  .... 
Uranus.  .  .  . 

Neptune  .  .  . 

388.  To  determine  the  position  of  the  nodes  of  a  planetary  orbit. — 
Let  the  longitude  of  a  planet  be  determined  when  it  is  at  one  of 
its  nodes ;  this  longitude  will  be  the  geocentric  longitude  of  the 
node.  Also,  by  means  of  the  solar  tables,  let  the  longitude  of  the 
sun  and  the  radius  vector  of  the  earth  be  found  for  the  time  the 
planet  is  at  the  node.  When  the  planet  returns  to  the  same  node 
again,  let  its  longitude  be  again  determined,  as  also  the  longitude 
of  the  sun  and  the  radius  vector  of  the  earth.  From  these  data 
(the  node  in  the  interval  being  supposed  to  remain  fixed)  the  po- 
sition of  the  line  of  the  nodes  may  be  determined,  and  also  the 
distance  of  the  planet  from  the  sun  at  the  times  of  observation. 

Let  S  be  the  place  of  the  sun,  E  the  earth,  and  P  a  superior 
planet  at  its  node ;  and  let  E'  be  the  place  of  the  earth  after  the 

O 


210 


ASTRONOMY. 


Fig.  103. 


planet  has  made  an  entire  revolution, 
and  returned  to  the  point  P.  Then 
from  the  solar  tables  we  can  determine 
(t  SE  and  SE',  as  also  the  angle  ESE'. 
Hence  EE'  can  be  computed,  as  also 
the  angles  SEE',  SE'B.  Now,  since 
the  angles  SEP,  SET  are  determined 
by  the  observations,  we  can  obtain  the 
angles  PEE',  PE'E.  Then,  in  the  tri- 
angle PEE',  having  two  angles  and  one 
side,  we  can  compute  PE.  Hence,  in 
the  triangle  PES,  we  have  two  sides 
and  the  included  angle,  from  which  we  can  compute  SP,  and  also 
the  angle  ESP,  which,  added  to  the  longitude  of  the  earth  when 
at  E,  will  give  the  heliocentric  longitude  of  the  planet  when  at  its 
node. 

When  observations  of  this  kind  are  made  at  a  considerable  dis- 
tance of  time  from  one  another,  it  is  found  that  the  nodes  of  every 
planet  have.a  slow  motion  retrograde,  or  in  a  direction  contrary 
to  the  order  of  the  signs.  The  most  rapid  motion  of  the  nodes  is 
in  the  case  of  Mercury,  amounting  to  about  70'  in  a  century. 

389.  To  determine  the  inclination  of  an  orbit  to  the  ecliptic. — Let 
the  time  at  which  the  sun's  longitude  is  the  same 
as  the  heliocentric  longitude  of  the  node  be  found 
by  means  of  the  solar  tables,  and  let  the  longitude 
and  latitude  of  the  planet  be  determined  at  the 
same  time. 

Let  NSE  be  the  line  of  a  planet's  nodes,  S  the 
sun,  E  the  earth,  and  P  the  planet's  place  in  its 
orbit.  From  E  as  a  centre,  with  a  radius  PE, 
suppose  a  sphere  to  be  described  whose  surface 
meets  the  line  NE  in  B ;  and  let  PA  be  an  arc  of 
a  great  circle  perpendicular  to  the  ecliptic.  Then  PBA  will  be 
a  spherical  triangle  right-angled  at  A;  the  angle  PBA  will 
measure  the  inclination  of  the  plane  of  the  planet's  orbit  to  the 
ecliptic ;  PA  will  measure  PEA,  the  geocentric  latitude  of  the 
planet;  and  AB  will  measure  AEB,  the  difference  between  the 
longitudes  of  the  sun  and  planet. 
Then,  by  Napier's  rule,  we  have 


ORBITS  .OF  THE   PLANETS.  211 

E  x  sin.  AB  =  tang.  PA  cot.  PB  A ; 


that  is,  tangent  of  inclination  equals  the  tangent  of  the  planet's  geocen- 
tric latitude,  divided  by  the  sine  of  the  difference  between  the  longitudes 
of  the  sun  and  planet,  the  earth  being  in  the  line  of  the  planet's 
nodes.  If,  at  the  time  of  observation,  the  elongation  of  the  plan- 
et from  the  sun  was  90°,  its  geocentric  latitude  would  be  the  incli- 
nation of  its  orbit  to  the  ecliptic;  and  the  results  of  this  method 
will  be  the  more  reliable  the  farther  the  planet  is  from  its  node. 

The  orbits  of  the  planets  have  generally  small  inclinations  to 
the  ecliptic.  The  orbit  of  Mercury  is  inclined  about  7°,  while  all 
the  other  planets  (with  the  exception  of  the  asteroids)  are  inclined 
less  than  4°.  Four  of  the  asteroids  have  inclinations  exceeding 
20°,  and  one  has  an  inclination  of  34°. 

390.  To  determine  the  heliocentric  longitude  and  latitude  of  a  planet. 
—  When  the  place  of  the  ascending  node  and  the  inclination  of 
the  orbit  of  a  planet  are  known,  the  heliocentric  longitude  and 
latitude  of  a  planet,  and  also  its  radius  vector,  may  be  deduced 
from  the  geocentric  longitude  and  latitude. 

Let  S  be  the  place  of  the  sun,  E  Fig  105. 

the  earth,  P  the  planet,  and  NS  the 
line  of  the  nodes  of  the  planet's  or- 
bit. From  P  draw  PB  perpendicu- 
lar to  the  ecliptic,  and  let  a  plane 
pnss  through  E,  P,  and  B,  intersect- 
ing the  line  of  the  nodes  in  N.  With 
N  as  a  centre,  and  NE  as  a  radius,  r 

let  a  sphere  be  described,  cutting  the  planes  PNS,  ENS,  and 
PNE  in  the  right-angled  spherical  triangle  AEG.  The  angle 
PEB  will  be  the  geocentric  latitude  of  the  planet,  BBS  will  be 
the  difference  between  the  longitudes  of  the  planet  and  sun,  and 
the  spherical  angle  ACE  will  measure  the  inclination  of  the 
planet's  orbit  to  the  ecliptic. 

1st.  In  the  triangle  NES,  the  angle  NES  is  known,  being  the 
supplement  of  BES  ;  also  ESN  can  be  derived  from  the  solar  ta- 
bles when  the  place  of  the  node  is  given,  and  ES  is  also  known  ; 
hence  we  can  compute  EN,  NS,  and  the  angle  ENS. 

2d.  In  the  spherical  triangle  AEC,  right-angled  at  E,  the  angle 


212 


ASTRONOMY. 


ACE  is  given,  and  also  EC,  which  measures  ENC ;  hence  AE, 
which  measures  ANE,  can  be  computed. 

3d.  In  the  triangle  PNE,  we  know  NE,  ENP,  and  NEP,  the 
supplement  of  the  planet's  geocentric  latitude ;  hence  PN  can  be 
computed. 

4th.  In  the  right-angled  triangle  NPB,  we  knpw  NP  and  the 
angle  PNB ;  hence  PB  and  NB  can  be  computed. 

5th.  In  the  triangle  BNS,  NB,  NS,  and  the  angle  BNS  are 
known ;  hence  we  can  compute  SB  and  NSB,  which  is  the  differ 
ence  between  the  heliocentric  longitude  of  the  planet  and  that  of 
its  node.     Hence  the  heliocentric  longitude  of  the  planet  is  d^j 
termined. 

6th.  In  the  right-angled  triangle  PBS,  we  know  PB  and  BS, 
from  which  we  can  compute  the  angle  PSB,  the  planet's  heliocen- 
tric latitude,  and  also  PS,  its  distance  from  the  sun. 

391.  To  determine  the  longitude  of  the  perihelion,  the  eccentricity, 
etc. — Assuming  the  orbit  of  the  planet  to  be  an  ellipse,  if  we  de- 
termine, by  Art.  390  or  Art.  388,  the  length  and  position  of  three 
radii  vectores  of  the  planet,  we  can  determine  the  form  and  di- 
mensions of  tbe  ellipse. 

Let  SB,  SC,  SD  be  three  radii  vec- 
tores of  the  planet,  given  in  length  and 
position.  Draw  the  lines  BC,  BD,  and 
produce  them,  making  SB  :  SD  : :  BF  : 
DF ;  and  SB  :  SC  : :  BE  :  CE ;  then 

SB-SD:  SB::  BD  :  BF  = 

and 

SB-SC:  SB  ::  BC:  BE  = 

IFK  H  G  Then  the  straight  line  passing  through 

the  points  E  and  F  will  be  the  directrix  of  the  ellipse.  For  BH, 
CI,  DK  being  drawn  perpendicular  to  EF,  the  triangles  BEH, 
CEI  are  similar ;  therefore  BH  :  CI : :  BE  :  CE.  Now,  by  con- 
struction, BE  :  CE  : :  SB  :  SC;  hence  BH  :  CI : :  SB  :  SC;  or  BH : 
SB  ::  CI :  SC;  also  BH  :  DK  : :  BF  :  DF  : :  SB  :  SD.  There- 
fore the  perpendiculars  BH,  CI,  DK  are  always  in  the  same  pro- 
portion as  the  lines  SB,  SC,  SD ;  consequently,  EF  is  the  directrix 
of  the  ellipse,  passing  through  B,  C,  and  D.  (Greom.,  Ellipse,  Prop. 


Fig.  100. 


THE  INFERIOR  PLANETS,  MERCURY  AND   VENUS.          213 

22.)  Through  S  draw  ASG  perpendicular  to  FE  ;  take  GA  :  AS 
::  CI  :  CS,and  GP  :  SP  ::  CI  :  CS;  then  CI+CS  :  CS  ::  GS  •  SP= 
SCxSG  ,  AC  SCxSG  . 

>  and  Ab=  ;  then  A  and  P  will  be  the  vertices 


of  the  ellipse. 

The  lengths  of  SP  and  SA  can  accordingly  be  computed  ;  their 
sum  gives  the  major  axis  ;  and  their  difference,  MS,  divided  by 
the  major  axis,  is  the  eccentricity  of  the  ellipse.  Also,  in  the  tri- 
angle BSM,  we  know  BS,  SM,  and  BM=PA-SB;  whence  the 
angle  BSA  is  determined,  which  gives  the  position  of  the  major 
axis  relatively  to  SB. 


CHAPTER  XV. 

THE  INFERIOR  PLANETS,  MERCURY  AND  VENUS. — TRANSITS. 

392.  Greatest  elongations  of  Mercury  and  Venus. — Mercury  and 
Venus  having  their  orbits  far  within  that  of  the  earth,  their  elon- 
gation or  angular  distance  from  the  sun  is  never  great.     They 
appear  to  accompany  the  sun,  being  seen  in  the  west  soon  after 
sunset,  or  in  the  east  a  little  before  sunrise. 

Let  S  be  the  place  of  the  sun,  MA  the  orbit  of 
Mercury,  E  the  place  of  the  earth,  and  M  the  place 
of  the  planet  when  at  its  greatest  elongation,  at 
which  time  the  angle  EMS  is  a  right  angle.  Since 
the  distances  of  the  planet  and  the  earth  from  the 
sun  both  vary,  the  greatest  elongation  must  also 
vary.  The  elongation  will  be  the  greatest  possi- 
ble when  SM  is  greatest  and  SE  is  the  least ;  that 
is,  when  Mercury  is  at  its  aphelion  and  the  earth 
at  perihelion.  Combining  the  greatest  value  of 
SM  with  the  least  value  of  SE,  we  find  the  greatest  possible  value 
of  Mercury's  greatest  elongation  to  be  28°  20';  but  in  the  present 
position  of  the  major  axis  of  Mercury's  orbit,  its  elongation  can 
not  exceed  27°  47',  and  the  least  value  of  its  greatest  elongation 
is  17°  51'. 

The  greatest  elongation  of  Venus  varies  from  45°  20'  to  47°  17". 

393.  Phases  of  Mercury  and  Venus. — The  planets  Mercury  and 


214  ASTRONOMY. 

Venus  exhibit  to  the  telescope  phases  similar  to  those  of  the 
moon.  At  the  greatest  elongations  eastward  or  westward,  we  see 
only  half  the  disc  illuminated,  as  in  the  case  of  our  own  satellite 

Fig.  108. 


at  first  or  last  quarter.  As  they  move  toward  the  superior  con- 
junction, at  A,  their  form  becomes  gibbous,  and  the  outline,  of  the 
disc  becomes  more  nearly  circular  the  nearer  they  approach  the 
superior  conjunction.  Owing  to  the  intensity  of  the  sun's  light, 
•we  lose  the  planets  for  a  little  time  before  and  after  the  conjunc- 
tion, but  on  emerging  from  the  sun's  rays  we  find  the  form  still 
gibbous.  The  illumined  part  diminishes  as  the  planets  approach 
their  greatest  elongation,  near  which  time  they  again  appear  as 
a  half  moon ;  and  as  they  advance  toward  the  inferior  conjunc- 
tion, the  form  becomes  more  nearly  that  of  a  crescent,  until  it  is 
again  lost  in  the  sun's  rays  at  C. 

MERCURY. 

394.  Period,  distance  from  sun,  etc. — Mercury  performs  its  revo- 
lution round  the  sun  in  a  little  less  than  three  months;  but  its  syn- 
odic period,  or  the  time  from  one  inferior  conjunction  to  another, 
is  116  days.  Its  mean  distance  from  the  sun  is  37  millions  of  miles. 

The  eccentricity  of  its  orbit  is  much  greater  than  in  the  case 
of  any  other  of  the  large  planets.  At  perihelion  Mercury  is  only 
29  millions  of  miles  from  the  sun,  while  in  aphelion  it  is  distant 
44  millions,  making  a  variation  of  15  millions  of  miles,  which  is 
about  one  fifth  of  the  major  axis  of  the  orbit. 

When  between  the  earth  and  the  sun,  the  disc  of  this  planet 
subtends  an  angle  of  about  twelve  seconds  of  arc ;  but  as  the 
planet  approaches  the  opposite  part  of  the  orbit,  its  breadth  does 
not  exceed  five  seconds.  The  real  diameter  of  Mercury  is  about 
3000  miles. 


THE   INFERIOR   PLANETS,  MERCURY  AND  VENUS.          215 

395.  Visibility  of  Mercury.  —  Since  tbe  elongation  of  Mercury 
from  tbe  sun  never  exceeds  27°  47',  this  planet  is  seldom  seen 
except  in  strong  twilight;  and  it  does  not  ever  appear  conspicu- 
ous to  the  naked  eye,  although  it  often  appears  as  brilliant  as  a 
star  of  the  first  magnitude  equally  near  to  the  sun.     Supposing 
the  atmosphere  to  be  clear,  the  other  circumstances  that  favor  the 
visibility  of  the  planet  are,  1st,  the  greatest  elongation  should  occur 
at  that  season  of  the  year  when  the  twilight  is  shortest;  2d,  the 
planet  should  then  be  near  the  aphelion  of  its  orbit;  and,  3rd,  the 
planet  should  be  near  its  greatest  distance  from  the  ecliptic  on 
the  north  side.     Unfortunately  the  position  of  Mercury's  orbit  is 
such  that  when  the  elongation  is  the  greatest  possible  the  planet 
is  south  of  the  sun,  and  the  circumstances  are  not  as  favorable  for 
observation  as  when  the  planet  has  a  less  elongation  but  is  north  of 
the  sun.     Generally  the  winter  months  afford  the  most  favorable 
opportunity  for  seeing  Mercury,  but  in  the  United  States  the  planet 
can  be  seen  by  the  naked  eye  at  any  season  of  the  year  provided 
the  observer  knows  exactly  where  and  when  to  look  for  it. 

396.  Greatest  brightness. — Mercury  does  not  appear  most  brill- 
iant when  near  superior  conjunction,  because  its  distance  is  then 
too  great;  nor  when  it  is  near  inferior  conjunction,  because  the 
illuminated  part  which  is  visible  to  us  is  then  very  small.     The 
place  of  greatest  brightness  must  lie  between  inferior  and  superior 
conjunction.     This  point  is  found  to  be  near  the  greatest  elonga- 
tion, and  is  between  the  greatest  elongation  and  superior  conjunc- 
tion.    When  the  planet  is  seen  after  sunset,  the  greatest  bright- 
ness occurs  a  few  days  before  the  greatest  elongation;  but  when 
it  is  seen  before  sunrise,  the  greatest  brightness  occurs  a  few  days 
after  the  greatest  elongation. 

397.  Rotation  on  its  axis. — Some  astronomers  think  they  have 
discovered  evidence  of  high  mountains  on  the  surface  of  Mercury, 
and  they  claim  to  have  seen  dark  streaks  and  spots  from  whose 
motion  they  have  concluded  that  the  planet  has  a  rotation  upon 
its  axis  in  about  24h.     But  these  observations  are  extremely  dif- 
ficult, and  the  conclusions  are  considered  doubtful. 

VENUS. 

398.  Yenus,  the  most  brilliant  of  the  planets,  is  generally  called 


216  ASTRONOMY. 

the  evening  or  the  morning  star.  The  evening  and  morning  star, 
or  the  Hesperus  and  Phosphorus  of  the  Greeks,  were  at  first  sup- 
posed to  be  different.  The  discovery  that  they  are  the  same  is 
ascribed  to  Pythagoras. 

399.  Period,  distance,  and  diameter. — Venus  revolves  round  the 
sun  in  about  7£  months ;  but  its  synodic  period,  or  the  time  from 
one  inferior  conjunction  to  another,  is  584  days,  or  about  19 
months.     Its  mean  distance  from  the  sun  is  69  millions  of  miles ; 
and  since  the  eccentricity  of  its  orbit  is  very  small,  this  distance 
is  subject  to  but  slight  variation. 

The  apparent  diameter  of  Venus  varies  much  more  sensibly 
than  that  of  Mercury,  owing  to  the  greater  variation  of  its  distance 
from  the  earth.  At  inferior  conjunction  its  disc  subtends  an  an- 
gle of  about  64  seconds  of  arc,  while  at  superior  conjunction  it  is 
less  than  10  seconds.  The  real  diameter  of  Venus  is  about  7700 
miles,  or  nearly  the  same  as  that  of  the  earth. 

400.  Venus  sometimes  visible  during  the  full  light  of  day. — The 
greatest  elongation  of  Venus  from  the  sun  amounts  to  47°,  and, 
on  account  of  its  proximity  to  the  earth,  it  is,  next  to  the  sun  and 
moon,  the  most  conspicuous  and  beautiful  object  in  the  firma- 
ment.   When  it  rises  before  the  sun,  it  is  called  the  morning  star ; 
when  it  sets  after  the  sun,  it  is  called  the  evening  star.     When 
most  brilliant,  it  can  be  distinctly  seen  at  midday  by  the  naked 
eye,  especially  if  at  the  time  it  is  near  its  greatest  north  latitude. 
Its  brightness  is  greatest  about  36  days  before  and  after  inferior 
conjunction,  its  elongation  being  then  about  40°,  and  the  enlight- 
ened part  of  the  disc  not  over  a  fourth  part  of  the  whole.     At 
these  periods  the  light  is  so  great  that  objects  illumined  by  it  at 
night  cast  perceptible  shadows. 

401.  Rotation  on  an  axis. — Astronomers  have  frequently  seen 
dusky  spots  upon  Venus,  which  have  been  watched  with  the  view 
of  ascertaining  the  time  of  a  rotation.     It  is  concluded  that  this 
time  is  about  23h.  21m. ;  but  these  observations  are  exceedingly 
difficult  on  account  of  the  glaring  light  of  the  planet. 

402.  Twilight  on  Venus. — By  observing  the  concave  edge  of  the 
crescent,  which  corresponds  to  the  boundary  of  the  illuminated 


TRANSITS  OF  MERCURY  AND  VENUS.  217 

and  dark  hemispheres,  it  is  found  that  there  is  a  gradual  fading 
away  of  the  light  into  the  darkness,  caused  probably  by  an  at- 
mosphere illuminated  by  the  sun  and  producing  the  phenomena 
of  twilight 

403.  Suspected  satellite. — Several  observers  of  the  last  two  cen- 
turies concurred  in  maintaining  that  they  had  seen  a  satellite  of 
Venus.     But  Sir  W.  Herschel  perceived  no  traces  of  a  satellite; 
neither  did  Schroter,  though  he  was  most  assiduous  in  his  obser- 
vations of  Venus.     It  is  therefore  probable  that  the  supposed  ap- 
pearances recorded  by  former  observers  were  illusive. 

TRANSITS  OF  MERCURY  AND  VENUS. 

404.  When  either  Mercury  or  Venus,  being  in  inferior  conjunc- 
tion, has  a  distance  from  the  ecliptic  less  than  the  sun's  semi-di- 
ameter, it  will  appear  projected  upon  the  sun's  disc  as  a  black 
round  spot.    The  apparent  motion  of  the  planet  being  then  retro- 
grade, it  will  appear  to  move  across  the  disc  of  the  sun  from  east 
to  west,  in  a  line  sensibly  parallel  to  the  ecliptic.     Such  a  phe- 
nomenon is  called  a  transit  of  the  planet. 

405.  When  transits  are  possible. — Transits  can  only  take  place 
when  the  planet  is  within  a  small  distance  of  its  node.    Let  1ST  be 
the  node  of  the  planet's  orbit ;  S  the  Fig.  100. 

centre  of  the  sun's  disc  on  the  eclip-  33"- 
tic,  and  at  such  a  distance  from  the 
node  that  the  edge  of  the  disc  just  touches  the  orbit,  NP,  of  the 
planet.  A  transit  can  only  take  place  when  the  sun's  centre  is 
at  a  less  distance  than  NS  from  the  node.  The  mean  value  of 
the  sun's  semi-diameter  being  16',  and  the  inclination  of  Mercury's 
orbit  to  the  ecliptic  being  7°,  and  that  of  Venus  3^-°,  we  find  that 
a  transit  of  Mercury  can  only  take  place  within  2°  11'  of  the 
node,  and  a  transit  of  Venus  within  4°  30'. 

406.  Transits  of  Mercury. — The  longitudes  of  Mercury's  nodes 
are  about  46°  and  226°,  at  which  points  the  earth  arrives  about 
the  10th  of  November  and  the  7th  of  May.     The  transits  of  Mer- 
cury must  therefore  occur  near  these  dates ;  those  at  the  ascend- 
ing node  taking  place  in  November,  and  those  at  the  descending 
node  in  May. 


218  ASTRONOMY. 

The  following  are  the  dates  of  the  transits  of  Mercury  for  the 
remainder  of  the  present  century : 


1868,  November  4. 
1878,  May  6. 
1881,  November  7. 


1891,  May  9. 
1894,  November  10. 


407.  Intervals  between  the  'transits. — In  each  of  these  cases  the 
interval  between  two  transits  at  the  same  node  is  13  years.     The 
reason  is  that  13  revolutions  of  the  earth  are  made  in  nearly  the 
same  time  as  54  revolutions  of  Mercury. 

For  365.256   x  13=4748.33. 

And  87.9692  x  54=4750.34 

When,  therefore,  a  transit  has  occurred  at  one  node,  after  an  in- 
terval of  13  years,  the  earth  and  Mercury  will  return  to  nearly  the 
same  relative  situation  in  the  heavens,  and  another  transit  may 
occur.  Transits  sometimes  occur  at  the  same  node  at  intervals 
of  7  years,  and  a  transit  at  either  node  is  generally  preceded  or 
followed,  at  an  interval  of  3£  years,  by  one  at  the  other  node. 

408.  Transits  of  Venus. — The  longitudes  of  the  nodes  of  Venus 
are  about  75°  and  255°,  at  which  points  the  earth  arrives  about 
the  5th  of  June  and  the  7th  of  December.     The  transits  of  Venus 
must  therefore  occur  near  these  dates ;  those  at  the  descending 
node  taking  place  in  June,  and  those  at  the  ascending  node  in 
^December. 

The  following  list  contains  all  the  transits  of  Venus,  from  that 
•which  took  place  in  1639  (the  first  that  was  ever  known  to  have 
been  seen  by  any  human  being)  to  the  end  of  the  present  century : 


1639,  December  4. 
1761,  June  5. 
1769,  June  3. 


1874,  December  8. 
1882,  December 


409.  Intervals  between  the  transits. — The  interval  between  two 
transits  at  the  same  node  is  either  8  or  235  years.  The  reason 
of  the  first  interval  is  that  8  revolutions  of  the  earth  are  accom- 
plished in  nearly  the  same  time  as  13  revolutions  of  Venus. 

For  365.256  x   8  =  2922.05. 

And  224.701  x  13  =  2921.11. 

Hence  a  transit  at  either  node  is  generally  preceded  or  followed, 
at  an  interval  of  8  years,  by  another  at  the  same  node. 


TRANSITS   OF   MERCURY  AND   VEXUS.  219 

The  period  of  235  years  is  still  more  remarkable. 

For  365.256x235=85835.3. 

And  224701x382=85835.7. 

Hence,  after  an  interval  of  235  years,  during  which  time  Venus 
has  made  382  revolutions,  the  earth  and  Venus  return  almost  ex- 
actly to  the  same  relative  situation  in  the  heavens. 

410.  Sun's  parallax  and  distance. — The  transits  of  Venus  are 
important  from  their  supplying  data  by  which  the  sun's  distance 
from  the  earth  can  be  determined  with  far  greater  precision  than 
by  any  other  known  method.  The  transits  of  Mercury  supply 
similar  data,  but  much  less  reliable,  on  account  of  the  greater  dis- 
tance of  that  planet  from  the  earth. 

The  relative  distances  of  the  planets  from  the  sun  may  be  com- 
puted by  Kepler's  third  law,  when  we  know  their  periods  of  rev- 
olution. In  this  manner  we  ascertain  that  the  distances  of  the 
earth  and  Venus  from  the  sun  are  in  the  ratio  of  1000  to  723. 
Hence,  when  Venus  is  interposed  between  the  earth  and  sun,  the 
ratio  of  its  distances  from  the  earth  and  sun  is  that  of  277  to 
723. 

Fig.  110. 


Let  the  circle  FHKG  represent  the  sun's  disc ;  let  E  represent 
the  earth,  and  A  and  B  the  places  of  two  observers  supposed  to 
be  situated  at  the  opposite  extremities  of  that  diameter  of  the 
earth  which  is  perpendicular  to  the  ecliptic;  also,  let  V  be  Venus 
moving  in  its  orbit  in  the  direction  represented  by  the  arrow.  At 
present  we  will  disregard  the  earth's  rotation  ;  that  is,  we  will 
suppose  the  positions  A  and  B  to  remain  fixed  during  the  transit 
The  planet  will  then  appear  to  the  observer  at  A  to  describe  the 
chord  FG,  and  to  the  observer  at  B  the  parallel  chord  HK.  Also, 
when  to  the  observer  at  A  the  centre  of  the  planet  appears  to  be 
at  D,  it  will  to  the  observer  at  B  appear  to  be  at  C. 

Now  AB  was  supposed  to  be  perpendicular  to  the  plane  of  the 
ecliptic ;  and  since  the  plane  of  the  sun's  disc  is  also  very  nearly 


220  ASTRONOMY. 

perpendicular  to  the  ecliptic,  the  line  AB  may  be  regarded  as  par- 
allel to  CD,  and  hence  we  have 

CD  :  AB  : :  DV :  AV : :  723  :  277  : :  2.61 :  1. 
Therefore         CD  (expressed  in  miles)  =  2.61  AB. 

The  apparent  distance  between  the  points  C  and  D  on  the  sun's 
surface  may  be  derived  from  the  observed  times  of  beginning  and 
ending  of  the  transit  at  A  and  B.     Let  the  observer  at  A  note 
Fig.  in.  the  time  when  the  disc  of  the  planet 

first  appears  to  touch  the  sun's  disc  on 
the  outside  at  L,  and  also  the  time  when 
it  first  appears  at  M  wholly  within  the 
sun's  disc.  L  is  called  the  external,  and 
M  the  internal  contact.  Also,  let  both 
the  internal  and  external  contacts  at  N 
and  P  be  observed  when  the  planet  is 
leaving  the  sun's  disc.  Then,  since  the 
planet's  rate  of  motion  as  well  as  that  of  the  sun  is  already  accu- 
rately known  from  the  tables,  the  number  of  seconds  of  a  degree 
in  the  chord  described  by  the  planet  can  be  ascertained.  In  the 
same  manner,  the  number  of  seconds  in  the  chord  described  by  the 
planet  as  observed  at  B  can  be  ascertained.  Knowing  the  length 
of  DG,  which  is  the  half  of  FG,  and  knowing  also  SG,  the  apparent 
radius  of  the  sun,  we  can  compute  SD.  In  the  same  manner,  from 
the  length  of  the  chord  HK,  we  can  compute  SC.  The  difference 
between  these  lines  is  the  value  of  CD,  supposed  to  be  expressed 
in  seconds.  But  we  have  already  ascertained  the  value  of  CD  in 
miles.  Hence  we  can  determine  the  linear  value  of  1"  at  the  sun 
as  seen  from  the  earth,  which  is  found  to  be  462  miles ;  and  hence 
the  angle  which  the  earth's  radius  subtends  at  the  sun  will  be 

3963 

^,  or  8".58.  This  angle  is  called  the  sun's  horizontal  parallax  ; 
462 

and  from  it,  when  we  know  the  radius  of  the  earth,  we  can  com- 
pute the  distance  of  the  earth  from  the  sun. 

It  is  not  necessary  that  the  observers  should  be  situated  at  the 
extremities  of  a  diameter  of  the  earth,  but  it  is  important  that 
the  two  stations  should  differ  widely  in  latitude;  and  allowance 
must  also  be  made  for  the  diurnal  motion  of  the  earth. 

The  transit  of  Venus  in  1769  was  observed  with  the  greatest 
care  at  a  large  number  of  stations,  extending  from  Lapland,  lati- 
tude 70°  22'  K,  to  Otaheite,  latitude  17°  25'  S.,  and  the  value  of 


THE  SUPERIOR  PLANETS.  221 

the  sun's  parallax  resulting  from  these  observations  (8".58)  is  that 
which,  until  recently,  has  generally  been  accepted  by  astronomers. 
The  mean  distance  of  the  earth  from  the  sun,  resulting  from 
this  value  of  the  sun's  parallax,  is  95,300,000  miles.  An  accu- 
rate knowledge  of  this  distance  is  of  the  greatest  importance, 
since  it  serves  as  our  base  line  for  estimating  the  distances  of  all 
bodies  situated  beyond  the  limits  of  our  solar  system.  See  Art. 
551.  As  there  is  still  some  uncertainty  respecting  the  exact 
value  of  this  quantity,  astronomers  generally  call  the  mean  dis- 
tance of  the  earth  from  the  sun  unity,  and  estimate  all  distances  in 
the  planetary  system  by  reference  to  this  unit. 

411.  Other  determinations  of  the  sun's  parallax. — When  Mars  is 
on  the  same  side  of  the  sun  with  the  earth,  it  approaches  com- 
paratively near  to  the  earth,  and  has  a  large  horizontal  parallax. 
Observations  on  the  position  of  Mars  have  repeatedly  been  made 
at  various  observatories,  both  in  the  northern  and  southern  hemi- 
spheres, from  which  the  parallax  of  this  planet  has  been  deduced; 
and  hence  the  parallax  of  the  sun  is  easily  computed,  since  the 
relative  distances  of  the  earth  and  Mars  from  the  sun  may  be  de- 
termined from  the  times  of  revolution.  The  horizontal  parallax  of 
the  sun  which  has  been  deduced  from  these  observations  is  8".8o. 

The  transit  of  Venus,  which  occurred  December  8, 1874,  was 
observed  with  the  greatest  care  at  a  large  number  of  stations  both 
in  the  northern  and  southern  hemispheres.  The  final  result, 
which  will  be  obtained  by  a  comparison  of  all  these  observations, 
is  not  yet  determined,  but  a  partial  comparison  indicates  a  par- 
allax of  S".S5.  There  can  be  no  doubt  that  the  value  of  the 
sun's  parallax  deduced  from  the  transit  of  Venus  in  1769  will 
require  to  be  somewhat  increased. 


CHAPTER  XVI. 

THE   SUPERIOR   PLANETS. — THEIR  SATELLITES. 

412.  How  the  superior  planets  are  distinguished  from  the  inferior. 
— The  superior  planets,  revolving  in  orbits  without  that  of  the 
earth,  never  come  between  us  and  the  sun — that  is,  they  have  no 
inferior  conjunction;  but  they  are  seen  in  superior  conjunction 
and  in  opposition.  Nor  do  they  exhibit  to  us  phases  like  those 


222  ASTRONOMY. 

of  Mercury  and  Ycnus.  The  disc  of  Mars,  about  the  period  of  hia 
quadratures,  appears  decidedly  gibbous ;  but  the  other  planets  are 
so  distant  that  their  enlightened  surfaces  are  always  turned  al- 
most entirely  toward  the  earth,  and  the  gibbous  form  is  not  per- 
ceptible. 

MARS. 

413.  Distance,  period,  etc. — The  mean  distance  of  Mars  from  the 
sun  is  145  millions  of  miles ;  but,  on  account  of  the  eccentricity 
of  its  orbit,  this  distance  is  subject  to  a  variation  of  nearly  one 
tenth  its  entire  amount.    Its  greatest  distance  from  the  sun  is  158 
millions  of  miles,  and  its  least  distance  132  millions. 

The  distance  of  this  planet  from  the  earth  at  opposition  is  some- 
times reduced  to  35  millions  of  miles,  while  at  conjunction  it  is 
sometimes  as  great  as  255  millions.  Its  apparent  diameter  varies 
in  the  same  ratio,  viz.,  3£"  to  24". 

Mars  makes  one  revolution  about  the  sun  in  687  days ;  but  its 
synodic  period,  or  the  interval  from  opposition  to  opposition,  is 
780  days.  The  inclination  of  its  orbit  to  the  plane  of  the  ecliptic 
is  1°  51'. 

The  real  diameter  of  this  planet  is  4500  miles,  and  its  volume 
about  one  fifth  that  of  the  earth. 

414.  Phases,  rotation,  etc.  —  At  opposition  and  conjunction,  the 
same  hemisphere  being  turned  to  the  earth  and  sun,  the  planet 

Fi&  112.          ^-— O-— -^  appears  like  a  full  moon,  as 

shown  at  Ml  and  M5.  In 
all  other  positions  it  appears 
slightly  gibbous ;  but  the  de- 
ficient portion  never  exceeds 
about  one  ninth  of  a  hemi- 
sphere. 

When  viewed  with  a  good 
telescope,  the  surface  of  Mars 
presents  outlines  of  what  are 
supposed  to  be  continents  and 
seas;  and  by  observing  these 
marks,  the  planet  has  been 

found  to  make  a  rotation  upon  its  axis  in  24h.  37m.,  and  its  axis 

is  inclined  to  the  axis  of  its  orbit  about  29°. 


MARS.  223 

Hence  the  days  and  nights  on  Mars  are  nearly  of  the  same 
length  as  on  the  earth;  the  year  is  diversified  by  seasons;  and 
the  surface  of  the  planet  by  climates,  not  very  different  from  those 
which  prevail  on  our  own  globe. 

415.  Spheroidal  form, — There  is  a  sensible  difference  between 
the  equatorial  and  polar  diameters  of  Mars,  amounting,  according 
to  some  astronomers,  to  one  fiftieth,  and,  according  to  others,  to 
one  thirty-ninth  of  the  equatorial  diameter.  This  is  much  greater 
than  corresponds  to  the  figure  of  equilibrium  of  a  liquid  planet 
making  one  rotation  in  24h.  37m. 

416..  Telescopic  appearance. — Many  of  the  spots  on  this  planet 
retain  the  same  forms,  with  the  same  varieties  of  light  and  shade, 
even  at  the  most  distant  intervals  of  time.  But  about  the  polar 
regions  are  sometimes  seen  white  spots,  with  a  well-defined  out- 
line, which  undergo  important  changes  from  one  season  to  anoth- 
er, and  which  may  be  explained  by  supposing  them  to  proceed 
from  polar  snows,  accumulated  during  the  long  winter,  and  which 
are  partially  dissolved  during  the  equally  protracted  summer. 

417.  Color. — Mars  exhibits  extensive  dusky  spots  of  a  dull  red 
hue,  which  are  believed  to  be  continents,  and  their  color  indicates 
a  reddish  tint  in  the  soil  like  the  red  sandstone  districts  of  the 
earth.    Other  portions  of  the  planet  are  of  a  greenish  hue,  and  are 
believed  to  be  tracts  of  water.     The  red  color  is  more  decided 
than  the  green,  and  gives  a  general  ruddy  appearance  to  the  disc. 

418.  Satellites. — In  1877  two  satellites  of  Mars  were  discovered 
at  the  Washington  Observatory.     The  distance  of  the  outer  sat- 
ellite from  the  centre  of  Mars  is  14,600  miles,  and  its  time  of 
revolution  is  30b.  18m.     The  distance  of  the  inner  satellite  is 
5850  miles,  and  its  time  of  revolution  is  7h.  39m.     Both  ob- 
jects are  extremely  faint,  and  are  probably  less  than  ten  miles  in 
diameter. 

419.  Sun's  parallax. — From  a  comparison  of  numerous  obser- 
vations of  Mars  when  near  opposition  in  1862,  the  parallax  of 
that  planet  was  determined,  and  hence  the  sun's  parallax  was 
computed  to  be  8".85,  which  is  a  little  greater  than  that  deduced 
from  the  transits  of  Venus. 


224:  ASTRONOMY. 

THE   MINOR  PLANETS,  OR  ASTEROIDS. 

420.  A  deficient  planei  between  Mars  and  Jupiter. — Nearly  three 
centuries  ago  Kepler  pointed  out  something  like  a  regular  pro- 
gression in  the  distances  of  the  planets  as  far  as  Mars,  which  was 
broken  in  the  case  of  Jupiter. 

In  1772,  Professor  Bode  announced  the  singular  relation  be- 
tween the  distances  of  the  planets  from  the  sun,  which  has  since 
been  known  as  Bode's  law.  This  law  is  as  follows :  If  we  set 
down  the  number  4  several  times  in  a  row,  and  to  the  second  4 
add  3,  to  the  third  4  add  twice  3  or  6,  to  the  next  4  add  twice  6 
or  12,  and  so  on,  the  resulting  numbers  will  represent  nearly  the 
relative  distances  of  the  planets  from  the  sun.  This  law  clearly 
indicated  a  deficient  planet  between  Mars  and  Jupiter;  and  an 
association  of  astronomers  was  formed  for  the  special  purpose  of 
searching  for  this  unknown  body. 

On  the  1st  of  January,  1801,  Piazzi  discovered  the  planet  Ceres, 
and  its  distance  was  found  to  correspond  very  nearly  with  that 
required  by  Bode's  law. 

In  1802,  Dr.  Olbers,  in  searching  for  Ceres,  discovered  another 
planet,  whose  orbit  was  found  to  have  nearly  the  same  dimen- 
sions as  that  of  Cerea  This  planet  was  called  Pallas. 

On  account  of  the  close  resemblance  in  appearance  between 
these  small  planets  and  the  fixed  stars,  Herschel  proposed  to  des- 
ignate them  by  the  name  Asteroid — a  term  which  has  been  very 
extensively  adopted.  Some  astronomers  employ  the  term  Plan- 
ttoid ;  but  the  term  minor  planet  is  more  descriptive,  and  is  now 
in  common  use  among  astronomers. 

421  Oilers' s  hypothesis  respecting  the  origin  of  the  asteroids. — Dr. 
Olbers  immediately  advanced  the  hypothesis  that  a  single  planet 
formerly  existed  between  Mars  and  Jupiter — that  it  was  broken 
into  fragments  by  volcanic  action  or  by  some  internal  force — 
that  Ceres  and  Pallas  were  two  of  its  fragments — and  that  prob- 
ably other  fragments  existed,  some  of  which  might  hereafter  be 
discovered. 

In  1804,  Professor  Harding  discovered  another  planet,  whose 
mean  distance  was  found  to  be  nearly  the  same  as  that  of  Ceres 
and  Pallas.  This  planet  was  named  Juno. 

In  1807,  Dr.  Olbers  discovered  still  another  planet,  whose  orbit 


THE   MINOR  PLANETS,  OR  ASTEROIDS.  225 

was  found  to  be  analogous  to  those  of  Ceres,  Pallas,  and  Juno. 
This  planet  was  named  Vesta. 

422.  Number  of  the  asteroids. — The  search  for  planets  was  pros- 
ecuted till  1816  without  farther  success,  when  it  was  discontin- 
ued ;  but  in  1845,  Hencke,  a  Prussian  observer,  having  resumed 
the  search,  discovered  another  small  planet,  which  has  been  named 
Astraea.     Since  that  time  the  progress  of  discovery  has  been  as- 
tonishingly rapid,  the  total  number  of  asteroids  known  in  1879 
amounting  to  209.   Of  these,  60  were  discovered  in  France,  50  in 
Germany,  19  in  Great  Britain  and  its  colonies,  67  in  America, 
and  11  in  Italy.     These  bodies  are  all  extremely  minute,  the 
largest  of  them  probably  not  exceeding  300  miles  in  diameter. 
vesta  is  the  only  one  among  them  which  is  ever  visible  to  the 
naked  eye,  and  this  only  under  the  most  favorable  circumstances. 

423.  Brightness  of  the  asteroids. — The  asteroids  closely  resemble 
small  stars,  and  can  only  be  distinguished  from  fixed  stars  by  their 
motion.     One  of  them,  when  near  the  opposition,  is  of  the  sixth 
magnitude ;  two  are  of  the  seventh  magnitude ;  five  of  the  eighth ; 
nineteen  of  the  ninth;  fifty-four  of  the  tenth;  seventy-seven  of  the 
eleventh;  forty-five  of  the  twelfth ;  and  five  of  the  thirteenth  mag- 
nitude.   Many  of  them  can  be  seen  only  near  the  opposition,  even 
by  the  largest  telescopes.    The  reason  that  no  asteroids  were  dis- 
covered for  so  long  a  period  after  1807  was  that  the  search  was 
conducted  with  too  little  system,  and  with  inadequate  instruments. 

424.  Distance  of  the  asteroids. — The  average  distance  of  the  209 
asteroids  from  the  sun  is  2.739,  or  260  millions  of  miles;  but  their 
distances  differ  widely  from  each  other.     The  asteroid  nearest  to 
the  sun  is  Flora,  with  a  mean  distance  of  209  millions  of  miles ; 
the  asteroid  most  remote  from  the  sun  is  Hilda,  with  a  mean  dis- 
tance of  375  millions  of  miles.     The  orbit  of  Flora  is  therefore 
nearer  to  that  of  Mars  than  to  that  of  Hilda. 

425.  Total  number  of  the  asteroids.— It  is  probable  that  there  is  a 
multitude  of  asteroids  yet  remaining  to  be  discovered.     From  an 
examination  of  the  influences  exerted  by  the  group  of  asteroids 
upon  the  planet  Mars,  Le  Verrier  has  concluded  that  the  entire 
mass  of  the  asteroids  between  Mars  and  Jupiter  may  amount  to 

P 


226  ASTRONOMY. 

one  third  part  of  the  mass  of  the  earth.  Now  it  would  require 
over  500  bodies  as  large  as  the  largest  of  the  asteroids  to  make  a 
body  one  third  of  the  size  of  the  earth ;  and,  since  many  of  the 
asteroids  are  extremely  minute,  their  number  probably  amounts 
to  many  thousands. 

426.  Is  Oilers 's  hypothesis  admissible  ?— The  hypothesis  of  01- 
bers  has  lost  most  of  its  plausibility  since  the  discovery  of  so 
many  asteroids.     If  these  bodies  ever  composed  a  single  planet, 
which  burst  into  fragments,  then,  since  the  orbits  all  started  from 
a  common  point,  each  must  return  to  the  same  point  in  every  rev- 
olution ;  in  other  words,  all  the  orbits  should  have  a  common 
point  of  intersection.     Such,  however,  is  far  from  being  the  case. 
The  orbits  are  spread  over  a  large  extent,  and  the  smallest  known 
orbit  is  every  where  distant  from  the  largest  by  at  least  50  mill- 
ions of  miles. 

427.  What  loas  the  origin  of  the  asteroid  system  ? — These  bodies, 
however,  exhibit  striking  resemblances,  which  point  to  some  pe- 
culiar relationship.    If  we  represent  all  the  orbits  under  the  form 
of  material  hoops,  or  rings,  these  rings  are  so  interlocked  as  to 
hang  together  as  one  system,  so  that  if  we  take  hold  of  any  one 
of  the  rings,  we  shall  lift  all  the  others  with  it.     This  feature  dis- 
tinguishes the  asteroid  orbits  from  all  the  other  orbits  of  the  solar 
system.    It  has  been  conjectured  that  all  the  planets  once  existed 
in  the  condition  of  gaseous  matter,  which  gradually  solidified  into 
spherical  masses.    If  such  were  the  case,  it  is  conceivable  that  the 
same  causes  which  determined  the  gaseous  matter,  once  occupy- 
ing an  immense  space  in  the  heavens,  to  collect  into  a  single  body 
and  form  a  large  planet,  like  Jupiter,  should,  in  another  part  of 
space,  have  produced  a  division  into  an  immense  number  of  small 
masses,  each  of  which  solidified  separate!}7,  thus  forming  the  group 
of  asteroids. 

428.  Are  there  asteroids  within  the  orbit  of  Mercury? — The  study 
of  the  motions  of  the  planet  Mercury  has  led  Le  Verrier  to  the 
conclusion  that  within  the  orbit  of  Mercury  there  exists  either  an 
undiscovered  planet,  whose  mass  is  nearly  equal  to  that  of  Mer- 
cury, or  else  a  ring  of  minute  planets  with  the  same  aggregate 
mass.     The  latter  supposition  is  regarded  as  the  most  probable. 


JUPITER.  227 

since  a  bright  planet  nearly  equal  in  size  to  that  of  Mercury  ought 
certainly  to  have  been  visible  during  total  eclipses  of  the  sun. 

JUPITER. 

429.  Distance,  period,  etc.— The  mean  distance  of  Jupiter  from 
the  sun  is  496  millions  of  miles ;  and,  since  the  eccentricity  of  its 
orbit  is  about  -^th,  this  distance  is  augmented  in  aphelion,  and 
diminished  in  perihelion  by  24  millions  of  miles.     On  account  of 
its  distance  from  the  sun  being  so  much  greater  than  that  of  the 
earth,  Jupiter  has  no  sensible  phases. 

Jupiter  makes  one  revolution  about  the  sun  in  11-g-  years;  and 
the  time  from  one  opposition  to  another  is  399  days. 

430.  Diameter. — Jupiter  is  the  largest  of  the  planets,  its  volume 
exceeding  the  sum  of  all  the  others.     Its  equatorial  diameter  is 
92,000  miles,  or  11  times  that  of  the  earth ;  and  its  volume  is 
1400  times  that  of  the  earth.     Its  apparent  diameter  varies  from 
30"  to  48".     When  near  opposition,  Jupiter  is  a  more  conspicu- 
ous object  in  the  heavens  than  any  other  planet  except  Venus, 
and  is  easily  seen  in  the  presence  of  a  strong  twilight 

431.  Rotation  on  an  axis,  spheroidal  form. — Permanent  marks 
have  been  occasionally  seen  on  Jupiter's  disc,  by  means  of  which 
its  rotation  has  been  distinctly  proved.     The  time  of  one  rotation 
is  9h.  55|m.     A  particle  at  the  equator  of  Jupiter  must  therefore 
move  with  a  velocity  of  more  than  450  miles  per  minute,  or  27 
times  as  fast  as  a  place  on  the  terrestrial  equator. 

The  Jovian  day  is  less  than  half  the  terrestrial  day ;  and  since 
the  period  of  Jupiter  is  4332  terrestrial  days,  it  consists  of  10,485 
Jovian  days. 

Jupiter's  equator  is  but  slightly  inclined  to  the  plane  of  its  or- 
bit, and  hence  the  difference  between  the  length  of  the  days  in 
summer  and  winter  is  very  small ;  and  the  change  of  temperature 
with  the  seasons  is  also  small. 

The  disc  of  Jupiter  is  oval,  the  polar  diameter  being  to  the  equa- 
torial as  16  to  17.  This  oblateness  is  found  by  computation  to  be 
the  same  as  would  be  produced  upon  a  liquid  globe,  making  one 
rotation  in  about  10  hours. 

432.  Belts  of  Jupiter.— When  viewed  with  a  good  telescope, 


228  ASTEONOMY. 

Jupiter's  disc  exhibits  a  light  yellowish  color,  having  a  series  of 
brownish-gray  streaks,  called  belts,  running  nearly  parallel  to  the 
equator  of  the  planet  Two  belts  are  generally  most  conspicu- 
ous, one  north  and  the  other  south  of  the  equator,  separated  by  a 
bright  yellow  zone.  These  belts  are  commonly  visible,  without 
much  change  of  form,  during  an  entire  rotation  of  the  planet.  Oc- 
casionally one  of  the  belts  appears  broken  sharply  off,  presenting 
an  extremity  so  well  defined  as  to  afford  the  means  of  determin- 
ing the  time  of  the  planet's  rotation. 

Near  the  poles  the  streaks  are  more  faint,  narrower,  and  less 
regular,  and  can  only  be  seen  with  good  telescopes.  All  the  belts 
become  less  distinct  toward  the  eastern  or  western  limb,  and  dis- 
appear altogether  at  the  limb  itself.  These  belts,  although  toler- 
ably permanent,  are  subject  to  slow  but  decisive  variations,  so 
that,  after  the  lapse  of  some  months,  the  appearance  of  the  disc  is 
totally  changed. 

433.  Cause  of  the  belts. — From  long-continued  observations,  it  is 
inferred  that  Jupiter  is  surrounded  by  an  atmosphere  which  is 
continually  charged  with  vast  masses  of  clouds,  which  almost  com- 
pletely conceal  the  surface  of  the  planet,  and  that  these  clouds 
have  a  permanence  of  form  and  position  much  greater  than  exists 
in  terrestrial  clouds. 

The  brightest  portion  of  Jupiter's  disc  probably  consists  of 
dense  clouds  which  reflect  the  light  of  the  sun,  while  the  darker 
spots  and  streaks  are  portions  of  the  atmosphere,  either  free  from 
clouds,  and  showing  the  surface  of  the  planet  more  or  less  dis- 
tinctly, or  they  are  clouds  of  inferior  reflecting  power. 

The  distribution  of  the  clouds  in  lines  parallel  to  the  equator 
is  probably  due  to  the  prevalence  of  atmospheric  currents,  analo- 
gous to  the  trade  winds,  and  arising  from  a  like  cause,  but  having 
a  constancy  and  intensity  far  greater  than  prevail  on  the  earth, 
on  account  of  the  more  rapid  rotation  and  greater  diameter  of 
Jupiter. 

434.  Jupiter's  satellites;  their  distances,  periods,  etc. — Jupiter  is 
attended  by  four  moons,  or  satellites,  revolving  around  the  pri- 
mary as  our  moon  revolves  around  the  earth,  but  with  a  much 
more  rapid  motion.     They  are  numbered  1,  2,  3, 4,  in  the  order 
of  their  distances  from  the  primary. 


JUPITEE.  229 

The  nearest  moon  completes  a  revolution  in  42  hours,  in  which 
time,  as  seen  from  Jupiter,  it  goes  through  all  the  phases  of  thin, 
crescent,  half  moon,  gibbous,  and  full  rnoon.  Its  distance  from 
Jupiter  is  280,000  miles.  The  distance  of  the  second  satellite  is 
440,000  miles,  and  it  completes  a  revolution  in  85  hours.  The 
distance  of  the  third  satellite  is  700,000  miles,  and  its  time  of 
revolution  172  hours.  The  distance  of  the  fourth  satellite  is 
1,200,000  miles,  and  its  time  of  revolution  is  400  hours,  or  16 
days  and  16  hours. 

These  satellites  were  discovered  by  Galileo,  at  Padua,  on  the 
8th  of  January,  1610.  When  viewed  with  a  telescope  of  moder- 
ate power,  they  present  the  appearance  of  minute  stars,  ranged 
nearly  in  the  direction  of  a  line  coinciding  with  the  planet's  equa- 
tor. Their  distances  from  the  primary  are  so  small  that  they  are 
all  .included  in  the  field  of  a  telescope  of  moderate  magnifying 
power,  the  distance  of  the  most  remote  one  being  only  13  times 
the  diameter  of  the  planet. 

The  real  diameter  of  the  smallest  satellite  is  2200  miles,  being 
the  same  as  the  diameter  of  our  moon  ;  and  the  diameter  of  the 
largest  satellite  is  3500  miles. . 

The  satellites  shine  with  the  brilliancy  of  stars  of  between  the 
sixth  and  seventh  magnitude ;  but,  owing  to  their  proximity  to 
the  planet,  which  overpowers  their  light,  they  are  in  general  in- 
visible without  the  aid  of  the  telescope.  On  high  mountains, 
where  the  air  is  extremely  rare,  they  have,  however,  been  detect- 
ed by  the  naked  eye. 

The  orbits  of  the  satellites  are  nearly  circular,  and  are  but  slight- 
ly inclined  to  the  plane  of  Jupiter's  orbit.  Hence  their  apparent 
motion  is  oscillatory,  going  alternately  from  their  greatest  elonga- 
tion on  one  side  to  the  greatest  elongation  on  the  other,  nearly  in 
a  straight  line. 

Oar  moon  makes  one  rotation  on  its  axis  in  the  same  time  that 
it  requires  to  revolve  around  the  earth.  It  is  thought  that  Ju- 
piter's moons  also  rotate  on  their  axes  in  the  time  of  their  respect- 
ive revolutions  round  the  planet.  This  is  inferred  from  period- 
ical fluctuations  in  the  brightness  of  the  satellites,  the  periods  cor- 
responding with  the  times  of  revolution  of  the  satellites. 

435.  Eclipses  of  the  satellites. — Jupiter's  satellites  frequently  pass 
into  the  shadow  of  the  primary,  and  become  invisible.  Let  JJ' 


230  ASTRONOMY. 

ns.  113. 


represent  the  planet  Jupiter ;  JVJ'  its  conical  shadow ;  SS'  the 
sun ;  E  and  E"  the  positions  of  the  earth  when  the  planet  is  in 
quadrature.  Let  ADFK  represent  the  orbit  of  one  of  the  satel- 
lites, whose  plane  we  will  suppose  to  coincide  with  the  ecliptic. 
From  E  draw  the  lines  EJ,  EJ',  meeting  the  path  of  the  satellite 
at  H  and  K,  as  also  at  C  and  D.  Let  A  and  B  be  the  points 
where  the  path  of  the  satellite  crosses  the  limits  of  the  shadow. 
By  a  computation  similar  to  that  employed  in  the  case  of  the 
earth,  Art  286,  we  find  that  the  length  of  Jupiter's  shadow  is 
more  than  50  millions  of  miles ;  and,  since  the  distance  of  the 
most  remote  satellite  is  but  little  over  one  million  miles,  the  sat- 
ellites pass  through  the  shadow  at  every  revolution.  In  extreme- 
ly rare  cases,  the  fourth  satellite,  on  account  of  the  inclination  of 
its  orbit  to  the  ecliptic,  passes  through  opposition  without  entering 
the  shadow. 

436.  Eclipses,  occultations,  transits,  etc. — In  the  revolution  of  the 
satellites  about  the  planet,  four  different  classes  of  phenomena  are 
observed : 

1st.  When  the  satellites  pass  into  the  shadow  of  the  planet  they 
are  said  to  be  eclipsed.  Their  entrance  into  the  shadow  at  A  is 
called  the  immersion  ;  their  passage  out  of  the  shadow  at  B  is 
called  the  emersion. 

2d.  When  the  satellites  pass  between  the  lines  SJ  and  S'J'  from 
F  to  G,  their  shadows  are  projected  on  the  surface  of  the  planet 
in  the  same  manner  as  the  shadow  of  the  moon  is  projected  on 
the  earth  in  a  solar  eclipse ;  and  in  this  case  the  shadow  may  be 
seen  moving  across  the  disc  of  the  planet  as  a  small  round  and 
black  spot.  This  is  called' a  transit  of  the  shadow. 

3d.  When  a  satellite,  passing  behind  the  planet,  is  between  the 
lines  EJC  and  EJ'D,  drawn  from  the  earth  at  E,  it  is  concealed 


JUPITER.  231 

from  the  observer  by  the  interposition  of  the  body  of  the  planet. 
It  disappears  on  one  side  of  the  planet's  disc,  and  reappears  on 
the  other.  This  phenomenon  is  called  an  occupation  of  the  satel- 
lite by  the  planet. 

4th.  When  a  satellite,  being  between  the  earth  and  planet,  passes 
between  the  lines  EJ  and  EJ',  drawn  from  the  earth  to  the  plan- 
et, its  disc  is  projected  on  that  of  the  planet ;  and  it  may  some- 
times be  seen  passing  across  the  disc,  being  brighter  or  darker 
than  the  ground  on  which  it  is  viewed,  according  as  it  is  projected 
on  a  dark  or  bright  belt.  This  is  called  a  transit  of  the  satellite. 
The  entrance  of  the  satellite  upon  the  disc  is  called  its  ingress,a.nd 
its  departure  is  called  its  egress. 

"When  the  planet  is  in  quadrature,  all  these  phenomena  may  be 
witnessed  in  the  revolution  of  the  satellites.  The  immersion  and 
emersion  of  the  third  and  fourth  satellites  at  A  and  B  may  both 
be  witnessed  on  the  same  side  of  the  planet  when  the  planet  is 
near  quadrature,  but  only  the  immersion  of  the  first  and  second 
satellites  is  visible.  The  view  of  their  emersion  is  intercepted  by 
the  body  of  the  planet,  and  they  do  not  reappear  until  after  hav- 
ing passed  behind  the  planet. 

437.  Longitude  determined  by  observations  of  the  eclipses. — The 
times  of  occurrence  of  all  these  phenomena  are  calculated  before- 
hand with  the  greatest  precision,  and  are  recorded  in  the  Nautical 
Almanac.     The  mean  time  of  their  occurrence  at  Greenwich  is 
there  given ;  so  that,  if  the  time  at  which  any  of  them  occur  at 
any  other  station  be  observed,  the  difference  between  the  local 
time  and  that  registered  in  the  Almanac  will  give  the  longitude 
of  the  place  from  the  meridian  of  Greenwich. 

This  method  of  determining  longitude  is,  however,  not  very  ac- 
curate ;  for,  since  the  light  of  a  satellite  decreases  gradually  while 
entering  the  shadow,  and  increases  gradually  on  leaving  it,  the 
observed  time  of  disappearance  or  reappearance  of  a  satellite  must 
depend  on  the  power  of  the  telescope  employed. 

438.  Configurations  of  the  satellites. — The  configurations  of  the 
satellites  of  Jupiter  are  continually  varying.    Sometimes  they  all 
appear  on  one  side  of  the  planet;  frequently  not  more  than  two 
or  three  of  the  satellites  are  visible ;  sometimes  only  one  satellite 
is  visible ;  and  a  few  instances  are  on  record  when  all  four  have 
been  invisible  for  a  short  time. 


232  ASTRONOMY. 

439.  Relation  of  the  mean  motions  of  the  first  three  satellites. — If 
the  mean  angular  velocity  of  the  first  satellite  be  added  to  twice 
that  of  the  third,  the  sum  will  be  equal  to  three  times  that  of  the 
second.     From  this  it  follows  that,  if  from  the  sum  of  the  mean 
longitude  of  the  first  and  twice  that  of  the  third,  three  times  that 
of  the  second  be  subtracted,  the  remainder  will  always  be  the 
same  quantity ;  and  from  observation  it  is  found  that  this  quan- 
tity is  180°.     Hence  it  also  follows  that  the  first  three  satellites 
can  never  all  be  eclipsed  at  once;  but  while  two  of  them  are 
eclipsed,  the  third  may  be  between  the  earth  and  Jupiter,  in  which 
position  a  satellite  is  often  entirely  invisible  unless  to  the  best  tel- 
escope. 

440.  Transmission  of  light. — Soon  after  the  invention  of  the  tel- 
escope, Roemer,  a  Danish  astronomer,  computed  a  table  showing 
the  time  of  occurrence  of  every  eclipse  of  the  satellites  of  Jupiter 
for  a  period  of  twelve  months.     He  then  observed  the  moments 
of  their  occurrence,  and  compared  his  observations  with  the  times 
registered  in  his  table.     At  the  commencement  of  his  observa- 
tions the  earth  was  at  E',  where  it  is  nearest  to  Jupiter.     As  the 
earth  moved  toward  E",  it  was  found  that  the  eclipses  occurred  a 
little  later  than  the  time  computed.     As  the  earth  moved  toward 
E'",  the  occurrence  of  the  eclipses  was  more  and  more  retarded, 
until  at  E'"  they  occurred  about  16  minutes  later  than  the  com- 
puted time.    While  the  earth  moved  from  E'"  to  E',  the  observed 
time  was  always  later  than  the  computed  time ;  but  this  differ- 
ence became  less  and  less,  until,  on  arriving  at  E',  the  observed 
time  agreed  exactly  with  the  computed  time. 

Thus  it  appeared  that  the  lateness  of  the  eclipse  depended  en- 
tirely upon  the  increased  distance  of  the  earth  from  Jupiter. 
When  the  earth  was  at  E"',  the  eclipse  was  observed  16  minutes 
later  than  when  the  earth  was  at  E' ;  and,  since  the  diameter  of 
the  earth's  orbit  is  190  millions  of  miles,  the  observation  of  the 
eclipse  was  delayed  one  second  for  every  200,000  miles  that  the 
earth's  distance  from  Jupiter  was  increased.  Now,  since  the 
eclipse  must  commence  as  soon  as  the  satellite  enters  Jupiter's 
shadow,  the  delay  in  the  observed  time  must  be  due  to  the  time 
required  for  the  light,  which  left  the  satellite  just  before  its  ex- 
tinction, to  reach  the  eye. 

By  more  exact  observations,  it  is  found  that  light  requires  16m. 


SATURN.  233 

26.6s.  in  crossing  the  earth's  orbit;  and  hence  the  velocity  of 
light  is  192,000  miles  per  second. 

SATURN. 

441.  Distance,  period,  etc. — The  mean  distance  of  Saturn  from 
the  sun  is  909  millions  of  miles ;  and,  on  account  of  the  eccen- 
tricity of  its  orbit,  this  distance  is  augmented  at  aphelion,  and  di- 
minished at  perihelion  by  more  than  -^th  of  its  whole  amount, 
varying  therefore  from  858  millions  to  960  millions  of  miles. 

Saturn  makes  one  revolution  about  the  sun  in  29£  years ;  and 
the  interval  between  two  successive  oppositions  is  378  days. 

442.  Diameter,  real  and  apparent. — Saturn  is  the  largest  of  all 
the  planets  except  Jupiter.     Its  equatorial  diameter  is  75,000 
miles,  being  more  than  nine  times  that  of  the  earth ;  and  its  vol- 
ume is  nearly  800  times  that  of  the  earth. 

The  mean  value  of  its  apparent  diameter  is  about  17";  and  it 
appears  as  a  star  of  the  first  magnitude,  with  a  faint  reddish  light. 
Its  disc  is  oval,  the  equatorial  diameter  being  TVth  greater  than 
the  polar.  The  disc  is  traversed  by  streaks  of  light  and  shade 
parallel  to  its  equator ;  but  these  belts  are  much  more  faint  than 
those  of  Jupiter.  These  belts  indicate  the  existence  of  an  atmos- 
phere surrounding  the  planet,  and  attended  with  the  same  system 
of  currents  which  prevail  on  Jupiter. 

443.  Rotation. — Saturn  makes  one  rotation  upon  its  axis  in  10-J 
hours ;  and  the  inclination  of  the  planet's  equator  to  the  plane 
of  the  ecliptic  is  28°.     Thus  the  year  of  Saturn  is  diversified  by 
the  same  succession  of  seasons  as  prevail  on  our  globe.    The  year 
in  Saturn  is  equal  to  10,700  terrestrial  days,  or  24,700  Saturnian 
days. 

444.  Saturn's  rings. — Saturn  is  surrounded  by  a  very  thin  plate 
of  matter  in  the  form  of  a  ring,  which  is  nearly  concentric  with 
the  planet,  and  in  the  plane  of  its  equator.     It  is  therefore  in- 
clined to  the  ecliptic  at  an  angle  of  28°,  and  intersects  it  in  two 
points,  which  are  called  the  ascending  and  descending  nodes  of 
the  ring.    With  powerful  telescopes  certain  dark  streaks  are  seen 
upon  its  surface,  bearing  some  resemblance  to  the  belts  of  the 
planet.     One  of  these  is  permanent  in  position,  and  indicates  that 


234  ASTRONOMY. 

the  ring  consists  of  two  concentric  rings  of  unequal  breadth,  one 
placed  outside  the  other,  without  any  mutual  contact. 

445.  Dimensions  of  the  rings. — The  distance  from  the  surface  of 
Saturn  to  the  inside  of  the  nearest  ring  is  about  19,000  miles ;  the 
breadth  of  this  ring  is  17,000  miles;  the  interval  between  the  two 
rings  is  1800  miles ;  and  the  breadth  of  the  exterior  ring  is  10,300 
miles.  The  greatest  diameter  of  the  outer  ring  is  172,000  miles. 
The  thickness  of  the  rings  is  extremely  small,  and  it  is  believed 
that  it  can  not  exceed  50  or  100  miles. 

4-16.  Varying  appearance  of  the  rings. — While  this  planet  moves 
in  its  orbit  round  the  sun,  the  plane  of  the  rings  is  carried  paral- 
lel to  itself,  so  that  during  a  revolution  it  undergoes  changes  of 
position  analogous  to  those  which  the  earth's  equator  exhibits. 
Twice  in  every  revolution — that  is,  at  intervals  of  15  years,  the 
plane  of  the  rings  must  pass  through  the  sun ;  and  the  ring,  if 
seen  at  all,  must  appear  as  a  straight  line.  As  the  planet  ad- 
vances in  its  orbit,  the  ring  appears  as  a  very  eccentric  ellipse. 
This  eccentricity  diminishes  until  Saturn  is  distant  90°  from  the 
nodes  of  the  ring,  when  the  minor  axis  of  the  ellipse  becomes 
equal  to  about  half  the  major  axis;  from  which  time  the  minor 
axis  decreases,  until,  at  the  end  of  half  a  revolution,  the  ring  again 
appears  as  a  straight  line. 

These  different  positions  of  Saturn's  ring  are  represented  in  the 
annexed  diagram,  where  S  represents  the  sun,  MN  the  orbit  of  the 


ISTB 


lau 


B 

earth,  and  A,  B,  C,  D,  etc.,  different  positions  of  Saturn.     "When 
Saturn,  is  at  A  and  E,  the  plane  of  the  ring  passes  through  the 


SATUEN. 


236 


sun,  and  only  the  edge  of  the  ring  can  be  seen,  as  represented  in 
the  figure ;  when  Saturn  arrives  at  B,  the  ring  appears  as  an  el- 
lipse ;  and  when  it  arrives  at  C,  the  minor  axis  of  the  ellipse  is 
equal  to  about  half  the  major  axis.  After  this  the  minor  axis  de- 
creases, and  when  the  planet  reaches  E  the  ring  appears  again  as 
a  straight  line. 

When  the  planet  is  in  quadrature,  a  portion  of  the  shadow 
which  it  projects  on  the  ring  is  visible  on  one  side  of  the  disc ; 
and,  in  certain  cases,  there  is  seen  a  portion  of  the  shadow  of  the 
ring  projected  on  the  planet's  disc.  Ttiese  phenomena  prove  that 
both  the  planet  and  the  ring  derive  their  illumination  from  the 
sun. 

447.  Disappearance  of  the  rings. — The  rings  of  Saturn  may  be- 
come invisible  from  the  earth  either  because  the  parts  turned  to- 
ward the  earth  are  not  illumined  by  the  sun,  or,  being  illumined, 
subtend  no  sensible  angle.  JFVrs£,When  the  plane  of  the  rings 
passes  through  the  sun,  only  the  edge  of  the  ring  is  illumined, 
and  this  is  too  thin  to  be  seen  by  any  but  the  most  powerful  tel- 
escopes. Second,  When  the  plane  of  the  rings  passes  through  the 
earth,  the  ring,  for  the  same  reason,  disappears  to  ordinary  tele- 
scopes. Third,  When  the  sun  and  the  earth  are  on  opposite  sides 
of  the  plane  of  the  rings — thnt  is,  when  the  plane  of  the  rings,  if 
produced,  passes  between  the  sun  and  the  earth,  the  dark  side  of 
the  rings  is  turned  toward  the  earth,  and  the  rings  entirely  dis- 
appear. 

Two  such  disappearances  usually  take  place  during  the  year  in 
which  the  plane  of  Saturn's  rings  crosses  the  earth's  orbit.  To 


Fijr.  115. 


show  this,  let  S  be  the  sun,  ABCD  the  earth's  orbit,  EFGr  a  part 
of  Saturn's  orbit,  and  F  the  position  of  Saturn  when  the  plane 


236  ASTRONOMY. 

of  the  rings,  if  produced,  would  pass  through  the  sun.  Draw  AG, 
CE,  parallel  to  SF,  touching  the  earth's  orbit  in  A  and  C.  Then, 
since  the  ring  always  preserves  its  parallelism,  its  plane  can  no- 
where intersect  the  earth's  orbit,  and  therefore  no  disappearance 
can  take  place,  unless  the  planet  be  between  E  and  G.  Now, 
since  SE,  the  distance  of  Saturn  from  the  sun,  is  to  SO,  that  of  the 
earth,  as  9.54  to  1,  the  angle  SEC  or  ESF  is  found  by  computa- 
tion equal  to  6°  1',  and  the  whole  angle  ESG-120  2';  and,  as 
Saturn's  periodic  time  in  his  orbit  is  10,759  days,  he  will  be  359^- 
days  in  describing  12°  2' — that  is,  about  6  days  less  than  a  com- 
plete year.  The  earth,  then,  describes  very  nearly  an  entire  rev- 
olution within  the  limits  of  time  when  a  disappearance  of  the  ring 
is  possible. 

448.  Number  and  duration  of  the  disappearances. — The  number 
of  these  disappearances  and  their  duration  will  depend  upon  the 
position  of  the  earth  in  its  orbit  when  the  planet  arrives  at  E. 

If,  when  Saturn  arrives  at  E,  the  earth  is  at  A,  the  earth  will 
encounter  the  plane  of  the  ring,  advancing  parallel  to  CE,  some- 
where in  the  quadrant  BC,  as  at  II.  The  ring  will  then  disappear, 
and  the  disappearance  will  continue  as  the  earth  proceeds  toward 
C,  because  the  dark  side  of  the  ring  is  toward  the  earth.  This 
disappearance  will  last  about  two  months,  and  close  when  the 
plane  of  the  ring  at  F  passes  the  sun,  for  after  that  time  the  il- 
lumined side  will  be  toward  the  earth.  While  the  earth  proceeds 
from  C  through  D  to  A,  the  plane  of  the  ring  will  move  from  FS 
to  GA,  and  will  pass  A  six  days  before  the  earth  reaches  that 
point.  In  this  case  there  will  be  but  one  period  of  disappearance 
of  the  ring,  lasting  about  two  months. 

If,  when  Saturn  is  at  E,  the  earth  is  at  K,  it  will  meet  and  pass 
through  the  advancing  plane  of  the  ring  somewhere  in  the  quad- 
rant BC,  after  which  the  dark  side  will  be  toward  the  earth. 
The  plane  of  the  ring  will  pass  the  sun  when  the  earth  is  on  the 
quadrant  CD,  after  which  the  bright  side  will  be  presented  to  the 
earth.  But  the  earth  will  overtake  the  nodal  line  before  it  reaches 
A,  and  therefore  look  again  upon  the  dark  side  until  it  recrosses 
the  line  somewhere  in  the  neighborhood  of  A.  Thus  there  will 
be  two  periods  of  disappearance.  These  two  periods  may  unite  in 
a  single  period  of  about  8  months'  duration,  and  this  will  happen 
when  the  earth  and  the  nodal  line  pass  D  at  the  same  instant ;  for 


SATURN.  237 

then  the  plane  of  the  ring  is  between  the  earth  and  sun  both  be- 
fore and  after  passing  the  point  D. 

If,  when  the  planet  is  at  F,  the  earth  is  at  B,  then  the  illumined 
side  of  the  ring  must  have  been  turned  toward  the  earth  during 
the  whole  time  that  the  planet  was  moving  from  E  to  F ;  and  the 
illumined  side  of  the  ring  will  also  be  turned  toward  the  earth 
during  the  whole  time  that  the  planet  moves  from  F  to  G ;  that 
is,  there  is  only  a  momentary  disappearance  of  the  ring ;  and 
even  this  can  never  be  observed,  because  the  planet,  being  in  con- 
junction with  the  sun,  is  lost  in  the  splendor  of  the  sun's  light. 

In  general,  during  the  year  in  which  the  line  of  the  ring's  nodes 
passes  the  earth,  there  are  two  periods  of  disappearance,  arising 
from  the  third  cause  mentioned  in  Art.  447,  each  beginning  and 
ending  with  a  disappearance  from  the  first  or  second  cause. 

449.  Observations  near  the  periods  of  disappearance. — The  last 
disappearance  of  Saturn's  ring  took  place  in  1862.     Observations 
near  these  periods  of  disappearance  have  indicated  the  existence 
of  great  inequalities  on  the  rings.     The  rings  frequently  present 
the  appearance  of  a  broken  line  of  light  projecting  from  each  side 
of  the  planet's  disc.     This  broken  appearance  may  be  explained 
by  supposing  inequalities  of  surface,  rendering  some  parts  of  the 
ring  so  thick  as  to  be  visible,  while  others  are  so  thin  as  to  be  in- 
visible when  presented  edgeways  to  the  observer.    It  is  probable, 
also,  that  the  rings  are  not  situated  exactly  in  the  same  plane. 

A  dark  line  has  sometimes  been  seen  dividing  the  outer  ring 
into  two,  which  seems  to  indicate  that  the  ring  is  really  triple ; 
and  some  observers  have  thought  that  they  had  discovered  evi- 
dence of  a  still  greater  number  of  divisions. 

450.  An  inner  ring  discovered  by  Professor  Bond. — In  1850,  Pro- 
fessor Bond,  at  Cambridge,  discovered  an  inner  ring,  composed 
of  matter  which  reflects  light  much  more  imperfectly  than  the 
planet  or  the  other  rings ;  and  is  transparent  to  such  a  degree 
that  the  body  of  the  planet  can  be  seen  through  it.     This  ring  is 
situated  between  the  planet  and  the  bright  rings,  and  approaches 
within  about  8000  miles  of  the  body  of  the  planet.    This  ring  has 
since  been  seen  by  numerous  observers  both  in  this  country  and 
in  Europe,  and  its  existence  is  unquestioned.     In  order  to  account 
for  the  fact  that  this  ring  has  never  been  seen  before,  it  has  been 


238  ASTRONOMY. 

conjectured  to  be  of  recent  formation.  It  appears  at  least  proba- 
ble that  this  ring  has  undergone  some  important  change  since  the 
time  of  Sir  William  Herschel. 

The  discovery  of  this  new  ring,  together  with  the  apparently 
variable  number  of  the  divisions  of  the  brighter  rings,  seem  to 
render  it  probable  that  the  rings  are  formed  by  a  cloud  of  sat- 
ellites too  small  to  be  separately  seen,  and  too  close  together  to 
allow  the  intervals  between  them  to  be  visible.  The  transparency 
of  the  inner  ring  may  be  explained  by  supposing  that  the  parti- 
cles are  so  scattered  that  we  can  see  through  the  cloud. 

451.  What  sustains  Saturn's  rings  f — Saturn's  rings  are  sustained 
in  precisely  the  same  manner  as  our  moon  is  sustained  in  its  rev- 
olution about  the  earth.    We  may  conceive  two  moons  to  revolve 
about  the  earth  in  the  same  orbit  as  the  present  one,  and  they 
would  be  sustained  by  the  same  law  of  attraction.     In  the  same 
manner,  three,  four,  or  a  hundred  moons  might  be  sustained.     In- 
deed, we  may  suppose  as  many  moons  arranged  around  the  earth 
as  would  complete  a  circle,  so  as  to  form  a  ring  of  moons  in  con- 
tact with  each  other.     They  would  all  be  sustained  in  the  same 
manner  as  our  present  moon  is  sustained.     If  we  conceive  these 
moons  to  be  cemented  together  by  cohesion,  we  shall  have  a  con- 
tinuous solid  ring;  and  the  ring  would  rotate  about  its  axis  in 
the  same  time  as  a  moon  situated  near  the  middle  of  its  breadth 
would  revolve  about  the  primary.     Observations  have  actually 
indicated  that  the  rings  of  Saturn  have  a  revolution  round  their 
common  centre,  and  in  their  own  plane,  in  a  period  of  lOh.  32m. 

452.  Appearance  of  the  rings  from  the  planet  itself. — The  rings  of 
Saturn  must  present  a  magnificent  spectacle  in  the  firmament  of 
that  planet,  appearing  as  vast  arches  spanning  the  sky  from  the 
eastern  to  the  western  horizon.    Their  appearance  varies  with  the 
position  of  an  observer  upon  the  planet.    To  an  observer  stationed 
at  Saturn's  equator,  the  ring  will  pass  through  the  zenith  at  right 
angles  to  the  meridian,  descending  to  the  horizon  at  the  east  and 
west  points.     If  the  observer  be  stationed  a  few  degrees  from  the 
equator,  on  the  same  side  of  the  ring  as  the  sun,  the  ring  will  pre- 
sent the  appearance  of  an  arch  in  the  heavens,  bearing  some  re- 
semblance in  form  to  a  rainbow.     If  we  suppose  the  observer  to 
travel  from  the  equator  toward  the  pole,  the  elevation  of  the  bow 


SATURN.  239 

will  diminish,  and  near  latitude  63°  it  will  descend  entirely  be- 
low the  horizon.  Beyond  this  parallel,  all  view  of  the  rings  will 
be  intercepted  by  the  convexity  of  the  planet.  Near  latitude  37° 
the  rings  are  seen  in  their  greatest  splendor,  forming  an  arch  15° 
in  breadth. 

453.  Saturn's  satellites;  their  distance, period,  etc. — Saturn  is  at- 
tended by  eight  satellites,  all  of  which,  except  the  most  distant 
one,  move  in  orbits  whose  planes  coincide  very  nearly  with  the 
plane  of  the  rings.  The  satellites  are  numbered  1, 2,  3,  etc.,  in  the 
order  of  their  distance  from  the  primar}r. 

The  sixth  satellite  is  the  largest,  and  was  first  discovered  by 
Iluygens  in  1655.  Its  distance  from  the  centre  of  the  planet  is 
778,000  miles,  and  the  time  of  one  revolution  is  about  16  days. 
Its  diameter  is  about  3000  miles.  It  shines  like  a  star  of  the 
eighth  magnitude,  and  in  powerful  telescopes  exhibits  a  decided 
disc. 

The  eighth  satellite  was  discovered  by  Cassini  in  1671.  Its  dis- 
tance from  the  centre  of  the  planet-  is  2,268,000  miles,  which  is 
nearly  twice  that  of  the  farthest  satellite  of  Jupiter,  and  the  time 
of  one  revolution  is  79  days.  Its  diameter  is  estimated  at  about 
1800  miles.  The  plane  of  its  orbit  is  inclined  10°  to  the  plane  of 
the  ring. 

Cassini  noticed  that  this  satellite  regularly  disappeared  during 
half  its  revolution  when  to  the  east  of  Saturn.  The  improvement 
of  telescopes  has  enabled  more  recent  observers  to  follow  the  sat- 
ellite through  the  entire  extent  of  its  orbit;  but  it  is  only  with 
the  greatest  difficulty  that  it  can  be  seen  on  the  eastern  side  of 
the  planet.  It  is  hence  inferred  that  this  satellite  rotates  on  its 
axis  in  the  time  of  one  revolution  round  the  primary ;  and  it  is 
probable  that  the  variations  in  its  brightness  are  owing  to  some 
parts  of  its  surface  being  less  capable  of  reflecting  the  sun's  light 
than  others.  At  maximum  brightness,  this  satellite  appears  like 
a  star  of  the  ninth  magnitude. 

The  fifth  satellite  was  discovered  by  Cassini  in  1672.  Its  dis- 
tance from  the  primary  is  336,000  miles,  and  its  period  of  revolu- 
tion 4J  days.  Its  diameter  is  estimated  at  1200  miles.  It  gener- 
ally shines  like  a  star  of  the  tenth  or  eleventh  magnitude. 

The  fourth  satellite  was  discovered  by  Cassini  in  1684.  Its 
distance  from  the  primary  is  240,000  miles,  and  its  period  of  rev- 


240  ASTRONOMY. 

olution  is  2f  days.  When  brightest,  it  appears  as  a  star  of  the 
eleventh  magnitude. 

The  third  satellite  was  discovered  by  Cassini  in  1684.  Its  dis- 
tance from  the  primary  is  188,000  miles,  and  its  period  is  1  day 
and  21  hours.  It  generally  resembles  a  star  of  the  thirteenth 
magnitude.  The  diameters  of  the  third  and  fourth  satellites  have 
been  estimated  at  500  miles. 

The  second  satellite  was  discovered  by  Sir  William  Ilerschel  in 
1787.  Its  distance  from  the  centre  of  the  primary  is  152,000 
miles,  and  its  period  of  revolution  is  1  day  9  hours.  It  appears 
as  a  star  of  the  fifteenth  magnitude. 

The  first  satellite  was  discovered  by  Sir  W.  Herschel  in  1789. 
Its  distance  from  the  centre  of  the  primary  is  118,000  miles,  and 
its  period  is  22£  hours.  This  satellite  describes  360°  of  its  orbit 
in  22£  hours,  being  at  the  rate  of  16°  per  hour.  Its  motion,  aa 
seen  from  the  primary,  must  therefore  be  so  rapid  as  to  resemble 
that  of  the  hour-hand  of  an  immense  time-piece.  In  two  minutes 
it  moves  over  a  space  equal  to  the  apparent  diameter  of  our  moon. 
This  satellite  is  an  extremely  faint  object,  and  can  only  be  seen 
by  the  largest  telescopes  under  the  most  favorable  circumstances. 

The  seventh  satellite  was  first  discovered  by  Professor  Bond, 
of  Cambridge,  September  16, 1848 ;  and,  two  days  later,  it  was 
seen  by  Mr.  Lassell,  of  Liverpool.  Its  distance  from  the  primary 
is  940,000  miles,  and  its  period  of  revolution  is  22  days.  It  re- 
sembles a  star  of  the  seventeenth  magnitude. 

454.  Mass  and  density  of  Saturn. — The  distance  of  a  satellite 
compared  with  its  time  of  revolution  enables  us  to  compare  the 
mass  of  Saturn,  or  its  quantity  of  matter,  with  that  of  the  earth. 
This  mass  is  thus  found  to  be  100  times  that  of  the  earth ;  but  its 
volume  is  nearly  800  times  that  of  the  earth ;  hence  its  density  is 
only  about  £th  that  of  the  earth.     Since  the  density  of  the  earth 
is  5£  times  greater  than  that  of  water,  the  density  of  Saturn  must 
be  about  £ ths  that  of  water.     This  is  the  density  of  the  lighter 
sorts  of  wood,  such  as  maple  and  cherry. 

URANUS.  ~ 

455.  Discovery. — In  1781,  the  attention  of  Sir  W.  Ilerschel  was 
attracted  to  an  object  which  he  did  not  find  registered  in  the  cata- 
logues of  stars,  and  which,  with  a  high  magnifying  power,  present- 


URANUS.  241 

ed  a  sensible  disc ;  and  he  soon  found  that  it  changed  its  place 
among  the  fixed  stars.  He  first  announced  this  object  as  a  comet ; 
but  when  it  was  found  to  move  in  an  orbit  nearly  circular  be- 
yond the  orbit  of  Saturn,  its  proper  place  among  the  planets  was 
no  longer  questioned,  and  it  was  proposed  to  call  it  the  "  Georgi- 
um  Sidus,"  in  compliment  to  George  III.  The  name  Herschel 
was  preferred  by  Laplace,  and  was,  to  some  extent,  adopted ;  but 
the  scientific  world  have  at  last  universally  agreed  upon  the  name 
Uranus. 

456.  Former  observations  of  this  planet. — As  soon  as  an  approxi- 
mate orbit  of  the  planet  had  been  obtained,  it  was  possible  to 
compute  its  place  at  any  past  epoch.     In  this  way  it  was  found 
that  the  planet  Uranus  had  been  observed  six  times  by  Flamsteed 
r.s  a  fixed  star,  twelve  times  by  Lemon nier,  and  once  by  Mayer. 
Thus  the  planet  had  been  observed  as  a  fixed  star  at  least  nine- 
teen times  before  its  real  nature  was  detected  by  Sir  W.  Herschel. 
These  observations  extend  back  to  1690,  and  have  proved  of  the 
greatest  value  in  accurately  determining  the  planet's  orbit. 

457.  Distance,  period,  etc. — The  mean  distance  of  Uranus  from 
the  sun  is  1828  millions  of  miles ;  and,  since  the  eccentricity  of  its 
orbit  is  very  small,  this  distance  is  increased  in  aphelion,  and  di- 
minished in  perihelion  by  less  than  one  twentieth  of  its  entire 
amount.     The  plane  of  its  orbit  coincides  nearly  with  that  of  the 
ecliptic. 

The  period  of  one  revolution  is  84  years ;  but  the  interval  be- 
tween two  successive  oppositions  is  only  370  days. 

458.  Diameter,  form  of  its  disc,  etc. — The  diameter  of  this  planet 
is  36,000  miles,  being  about  half  that  of  Saturn,  and  more  than 
four  times  that  of  the  earth.     Its  volume  is  nearly  100  times  that 
of  the  earth  ;  and  its  apparent  diameter  is  about  4". 

The  planet  may  be  just  discerned  by  a  person  gifted  with 
strong  sight,  without  the  telescope,  in  a  perfectly  dark  sky,  when 
its  exact  position  with  reference  to  the  surrounding  stars  is  known. 

The  disc  of  Uranus  appears  uniformly  bright,  and  of  a  pale 
color,  but  no  appearance  of  spots  or  belts  has  been  perceived. 
For  this  reason,  the  time  of  rotation  upon  its  axis  has  not  been 
ascertained.  Some  astronomers  think  they  have  detected  consid- 

Q 


242  ASTRONOMY. 

erable  ellipticity  in  the  form  of  the  planet;  but  other  astronomers, 
with  equally  good  telescopes,  have  not  succeeded  in  discovering 
any  difference  in  the  diameters. 

Since  light  moves  at  the  rate  of  192,000  miles  per  second,  it 
would  require  over  9000  seconds,  or  2£  hours,  to  move  from  the 
sun  to  Uranus.  Whatever  changes  may  take  place  on  the  sur- 
face of  the  sun,  they  can  not,  therefore,  be  perceived  by  inhab- 
itants of  that  planet  until  2£  hours  after  they  really  take  place. 

459.  Satellites. — Soon  after  the  discovery  of  this  planet,  Sir  W. 
Herschel  announced  that  it  was  attended  by  a  system  of  six  sat- 
ellites,  but  only  four  have  ever  been  seen  by  any  other  observer. 
The  times  of  revolution  of  these  four  satellites,  together  with  their 
distances  from  the  primary,  have  been  well  determined,  and  are 
as  follows: 


Satellite. 

Revolution. 

Distance. 

1 

2 
3 

4 

2  d.    12  h. 
4         3 
8       17 
13       11 

120,000  miles. 
171,000     " 
288,000     " 
380,000     " 

The  third  and  fourth  of  these  satellites  are  by  far  the  most  con- 
spicuous, and  their  periods  have  been  ascertained  with  great  ac- 
curacy. The  existence  of  the  fifth  and  sixth  satellites  announced 
by  Herschel  must  be  regarded  as  quite  doubtful. 

When  the  plane  of  the  orbits  of  the  satellites  passes  through 
the  earth,  the  orbits  appear  as  straight  lines.  Such  was  the  case 
in  1840.  When  the  direction  of  the  earth  is  at  right  angles  to 
the  line  of  the  nodes,  the  apparent  orbits  do  not  differ  sensibly 
from  circles.  Such  was  the  case  in  1862.  In  all  intermediate 
positions,  the  apparent  orbits  are  more  or  less  elliptical. 

Contrary  to  the  law  which  generally  prevails  in  the  motions  of 
the  planets  and  their  satellites,  the  orbits  of  these  satellites  are  in- 
clined to  the  plane  of  the  ecliptic  79°,  being  little  less  than  a  right 
angle ;  and  their  motions  in  these  orbits  are  retrograde — that  is, 
from  east  to  west.  We  may,  however,  consider  the  motion  of  the 
satellites  as  direct,  in  orbits  inclined  101°  to  the  plane  of  the  eclip- 
tic. 


NEPTUNE.  243 


NEPTUNE. 

460.  Perturbations  of  the  planets. — If  the  planets  were  subject 
only  to  the  attraction  of  the  sun,  they  would  revolve  in  exact  el- 
lipses, of  which  the  sun  would  be  the  common  focus ;  but,  since 
they  arc  also  subject  to  the  attraction  of  each  other,  they  are  drawn 
slightly  out  of  the  ellipses  which  they  would  otherwise  describe. 
When  the  masses  and  distances  of  the  planets  are  known,  these 
disturbances  can  be  computed  with  such  precision  that  the  exact 
place  of  any  planet  can  be  determined  for  any  time  either  past  or 
future. 

461.  Irregularities  in  the  motion  of  Uranus.-^-In.  1821,Bouvard 
published  a  set  of  tables  for  computing  the  place  of  Uranus.    The 
materials  for  the  construction  of  these  tables  consisted  of  40  years' 
regular  observations  since  1781,  and  the  19  accidental  observa- 
tions (Art.  456),  reaching  back  almost  a  century  farther.    Bouvard 
was  unable  to  find  any  elliptic  orbit  which,  combined  with  the 
perturbations  of  known  planets,  would  represent  the  entire  series 
of  observations.     He  therefore  rejected  the  ancient  observations, 
and  founded  his  tables  upon  the  observations  since  1781.     These 
tables  represent  tolerably  well  the  observations  of  the  40  years 
from  which  they  were  derived;  but  they  do  not  represent  the 
observations  of  the  planet  made  before  1781,  nor  do  they  repre- 
sent the  observations  made  since  1820.     From  1690  to  1715  the 
observed  place  of  the  planet  was  considerably  in  advance  of  its 
computed  place ;  from  1715  to  1771  it  was  considerably  behind ; 
and  since  1825  it  has  been  behind  its  computed  place,  and  the 
error  has  been  rapidly  increasing.     In  1830  the  deviation  of  the 
planet  from  its  computed  place  amounted  to  twenty  seconds  of 
arc;  in  1840  the  deviation  amounted  to  ninety  seconds;  and  in 
1844  it  amounted  to  one  hundred  and  twenty  seconds.     These 
discrepancies  are  far  too  great  to  be  ascribed  to  the  inaccuracy 
of  the  observations,  and  we  must  conclude  that  they  result  from 
some  cause  of  which  Bouvard  did  not  take  account  in  the  con- 
struction of  his  tables. 

In  order  to  explain  these  anomalies,  it  was  conjectured  by 
some  astronomers  that  the  Newtonian  law  of  gravitation  was 
not  rigorously  exact  at  the  distance  of  Uranus,  while  others 
supposed  that  the  discrepancies  might  be  accounted  for  by 


244  ASTEONOMY. 

the  action  of  an  unknown  planet  situated  beyond  the  orbit  of 
Uranus. 

462.  Researches  of  Le  Verrier  and  Adams. — In  the  year  1845,  two 
astronomers,  M.  Le  Verrier,  of  Paris,  and  Mr.  Adams,  of  Cambridge, 
England,  independently  of  each  other,  attempted  to  determine  the 
place  and  magnitude  of  a  planet  outside  of  Uranus,  which  would 
account  for  these  irregularities.     The  problem  which  they  pro- 
posed, and  which  they  actually  solved,  was  this:  Given  the  pertur- 
bations produced  in  Uranus  by  the  action  of  an  unknown  planet ;  it 
is  required  to  assign  the  elements  of  a  planet  capable  of  producing 
these  perturbations. 

Le  Verrier  and  Adams,  by  a  most  laborious  analysis,  demon- 
strated that  these  irregularities  were  such  as  would  be  caused  by 
an  undiscovered  planet  revolving  about  the  sun  at  a  distance 
nearly  double  that  of  Uranus,  and  with  a  mass  somewhat  greater 
than  that  of  Uranus;  and  they  pointed  out  the  place  in  the  heav- 
ens which  this  planet  ought  at  present  to  occupy.  Le  Verrier 
was  the  first  to  publish  to  the  world  the  results  of  his  research- 
es, and  thus  obtained  the  chief  credit  for  the  discovery. 

463.  Discovery  of  the  planet  at  Berlin.—  On  the  23d  of  September, 
1846,  Dr.Galle,  of  the  Berlin  Observatory,  received  a  letter  from 
Le  Verrier,  announcing  the  results  of  his  calculations,  informing 
him  that  the  longitude  of  the  unseen  planet  ought  to  be  326°, 
and  requesting  him  to  search  for  it.     Dr.  Galle  did  search  for  it, 
and  found  it  on  the  first  night.    It  appeared  as  a  star  of  the  eighth 
magnitude,  having  a  longitude  of  326°  52',  and,  consequently,  only 
52'  from  the  place  assigned  by  Le  Verrier.     This  planet  has  been 
called  Neptune. 

The  orbit  of  Neptune  is  smaller  than  that  predicted  by  either 
Adams  or  Le  Verrier,  and  its  mass  somewhat  less;  yet  its  disturb- 
ing action  upon  Uranus  is  such  as  perfectly  to  explain  the  anom- 
alies which  had  been  observed  in  the  motion  of  that  planet 

464.  Earlier  observations  of  this  planet. — As  soon  as  an  approx- 
imate orbit  of  Neptune  had  been  obtained,  its  place  was  computed 
back  for  several  preceding  years,  and  it  was  found  that  it  had  been 
repeatedly  observed  as  a  fixed  star.    Two  such  observations  were 
made  in  1795,  one  in  1845,  and  three  in  1846,  before  it  was  seen 


NEPTUNE. 


245 


at  Berlin.  With  the  aid  of  these  observations,  it  was  soon  possi- 
ble to  obtain  a  very  accurate  determination  of  the  orbit  of  the 
planet. 

465.  Distance,  period,  etc. — The  mean  distance  of  Neptune  from 
the  sun  is  2862  millions  of  miles,  and  its  period  of  revolution  is 
164  years.     Its  apparent  diameter  is  about  2|  seconds,  and  it  re- 
sembles a  star  of  the  eighth  magnitude.     Its  real  diameter  is 
35,000  miles,  which  is  a  little  less  than  that  of  Uranus. 

466.  Bode's  law  disproved. — The  discovery  of  Neptune  has  en- 
tirely refuted  Bode's  law  of  planetary  distances.     This  law  has 
been  stated  in  Art  420.     The  following  table  shows,  first,  the  true 
relative  distance  of  each  of  the  planets ;  second,  the  distance  ac- 
cording to  Bode's  law ;  and,  third,  the  error  of  this  law. 


True  Distance. 

Bode's  Law. 

Error. 

Mercury     .     .     . 

8.87 

4 

0.13 

Venus   .... 

7.23 

7 

0.23 

Earth    .... 

10.00 

10 

Mars     .... 

15.24 

16 

0.76 

200  Asteroids      . 

27.39 

28 

0.61 

Jupiter  .... 

52.03 

52 

0.03 

Saturn  .... 

95.39 

100 

4.61 

Uranus  .... 

191.82 

196 

4.18 

Neptune 

300.37 

388 

87.63 

Hence  it  will  be  seen  that,  although  this  law  represents  pretty 
well  the  distances  of  the  nearer  planets,  the  error  is  quite  large 
for  Saturn  and  Uranus,  and  for  Neptune  the  error  amounts  to 
more  than  800  millions  of  miles. 

467.  Satellite  of  Neptune.  —  Neptune  has  one  satellite,  which 
makes  a  revolution  around  the  primary  in  5d.  21b.,  at  a  distance 
of  236,000  miles,  which  is  about  the  same  as  the  distance  of  our 
moon  from  the  earth.  The  orbit  of  this  satellite  is  inclined  29°  to 
the  plane  of  the  ecliptic,  and  its  motion  in  this  plane  is  retrograde. 
This  fact  is  one  of  great  interest,  as  hitherto  the  only  known  in- 
stance of  retrograde  motion  among  the  planets,  or  their  satellites, 
has  been  the  case  of  the  satellites  of  Uranus.  This  satellite  is  es- 
timated to  be  equal  in  brightness  to  a  star  of  the  fourteenth  mag- 
nitude. 


24.6  ASTRONOMY. 

Several  observers  at  first  suspected  that  Neptune  was  attended 
by  a  ring  like  Saturn,  but  later  observations  do  not  countenance 
this  idea. 

468.  Appearance  of  the  solar  system  as  observed  from  Neptune. — 
The  apparent  diameter  of  the  sun  as  seen  from  Neptune  is  64", 
which  but  little  exceeds  the  greatest  apparent  diameter  of  Venus 
as  seen  from  the  earth.  The  illuminating  effect  of  the  sun  at  that 
distance  is  only  about  one  thousandth  part  of  its  effect  upon  the 
earth,  being  about  midway  between  our  sunlight  and  our  moon- 
light 

With  reference  to  Neptune,  all  the  other  planets  are  inferior, 
and  most  of  them  never  appear  to  recede  many  degrees  from  the 
sun.  The  greatest  elongation  of  Uranus  is  40°,  of  Saturn  18°,  of 
Jupiter  10°,  of  Mars  3°,  and  of  the  interior  planets  still  less.  Ura- 
nus, Saturn,  and  Jupiter  might  perhaps  therefore  be  seen  by  the 
inhabitants  of  Neptune  as  stars  of  the  sixth  magnitude,  but  none 
of  the  remaining  planets.  All  the  planets,  if  they  could  be  ob- 
served from  Neptune,  would  occasionally  appear  to  travel  across 
the  sun's  disc,  but  those  which  are  interior  to  Jupiter  subtend  so 
small  an  angle  that  it  is  doubtful  whether  they  could  be  seen  even 
with  the  best  telescope ;  and,  on  account  of  the  small  diameter  of 
the  sun,  combined  with  the  inclination  of  the  planetary  orbits,  the 
transits  of  the  larger  planets  would  be  of  extremely  rare  occur- 
rence. A  transit  of  Uranus  would  not  happen  oftener  than  once 
in  40,000  of  our  years. 

The  problem  of  finding  the  distance  of  the  fixed  stars  presents 
very  little  difficulty  to  the  Neptunian  astronomers,  except  that 
which  arises  from  the  length  of  one  of  their  years,  required  to  com- 
plete an  observation,  since  they  are  in  possession  of  a  base  lino 
thirty  times  as  long  as  that  to  which  we  are  confined.  See  Art 
ool. 


QUANTITY  OF  MATTER  IN  THE  SUN  AND  PLANETS.       247 


CHAPTER  XVII. 

QUANTITY  OF  MATTER  IN  THE   SUN  AND  PLANETS.  —  PLANETARY 
PERTURBATIONS. 

469.  How  to  determine  the  mass  of  a  planet.  —  By  the  method  em- 
ployed in  Art.  266,  we  may  determine  the  masses  of  such  of  the 
planets  as  have  satellites.  The  quantity  of  matter  may  also  be 
found  in  terms  of  the  distance  and  periodic  time  of  the  planet  and 
its  satellite. 

Let  M  represent  the  mass  of  the  sun,  R  the  distance  of  a  planet, 
and  T  its  periodic  time  ;  then,  by  Art.  248,  the  central  force  which 
retains  the  planet  in  its  orbit  is 


T2   ' 

But,  since  the  planet  is  retained  in  its  orbit  by  the  attraction 
of  the  sun,  and  this  attraction  varies  directly  as  the  mass,  and  in- 
versely as  the  square  of  the  distance,  Art.  256,  we  shall  have 


or  M=^~-. 

For  the  same  reason,  if  we  put  m  to  represent  the  mass  of  a 
planet,  r  the  distance,  and  t  the  periodic  time  of  a  satellite  revolv- 
ing around  it,  we  shall  have 

47rV  /9x 

m=-f-.  (2) 

Comparing  equations  (1)  and  (2),  we  find 

...  R3    r3 

M:  m::  ^:  — . 

Hence  we  see  that  the  quantities  of  matter  in  the  bodies  -which 
compose  the  solar  system  are  directly  as  the  cubes  of  the  mean 
distances  of  any  bodies  which  revolve  about  them,  and  inversely 
as  the  squares  of  the  times  in  which  the  revolutions  are  per- 
formed. 

Ex,  1.  The  distance  of  the  earth  from  the  sun  is  95,300,000 


248  ASTRONOMY. 

miles,  and  its  time  of  revolution  365.256  days.  The  distance  of 
the  moon  from  the  earth  is  238,900  miles,  and  its  time  of  revolu- 
tion 27.321  days.  What  is  the  mass  of  the  sun  compared  with 
that  of  the  earth  ?  Ans.  355,000  times  that  of  the  earth. 

Ex.  2.  The  mean  distance  of  Jupiter  from  the  sun  is  495,817,000 
miles,  and  its  time  of  revolution  is  103,982  hours;  the  distance 
of  its  fourth  satellite  is  1,200,000  miles,  and  its  time  of  revolution 
400.53  hours.  What  is  the  mass  of  the  sun  compared  with  that 
of  Jupiter?  Ans.  1047  times  that  of  Jupiter. 

Ex.  3.  What  is  the  mass  of  Jupiter  compared  with  that  of  the 
earth  ?  Ans.  339  times  that  of  the  earth. 

Ex  A.  The  distance  of  Saturn  from  the  sun  is  909,028,000  miles, 
and  its  time  of  revolution  10,759  days;  the  distance  of  its  outer 
satellite  is  2,268,000  miles,  and  its  time  of  revolution  79.32  days. 
What  is  the  mass  of  the  sun  compared  with  that  of  Saturn  ? 

Ans.  3500  times  that  of  Saturn. 

Ex.  5.  What  is  the  mass  of  Saturn  compared  with  that  of  the 
earth?  Ans.  101  times  that  of  the  earth. 

Ex.  6.  The  distance  of  Uranus  from  the  sun  is  1,828,200,000 
miles,  and  its  time  of  revolution  30686.8  days ;  the  distance  of  its 
fourth  satellite  is  380,000  miles,  and  its  time  of  revolution  13.463 
days.  What  is  the  mass  of  the  sun  compared  with  that  of  Ura- 
nus ?  Ans.  21,400  times  that  of  Uranus. 

Ex.  7.  What  is  the  mass  of  Uranus  compared  with  that  of  the 
earth  ?  Ans.  16  times  that  of  the  earth. 

Ex.  8.  The  distance  of  Neptune  from  the  sun  is  2,862,457,000 
miles,  and  its  time  of  revolution  60126.7  days ;  the  distance  of  its 
satellite  is  236,000  miles,  and  its  time  of  revolution  5.87  days. 
What  is  the  mass  of  the  sun  compared  with  that  of  Neptune  ? 
Ans.  17,000  times  that  of  Neptune. 

Ex.  9.  What  is  the  mass  of  Neptune  compared  with  that  of  the 
earth  ?  Ans.  21  times  that  of  the  earth. 

The  masses  of  those  planets  which  have  no  satellites  have  been 
determined  by  estimating  the  force  of  attraction  which  they  exert 
in  disturbing  the  motions  of  other  bodies.  The  mass  of  Mercury- 
has  been  determined  from  the  perturbations  which  it  causes  in 
the  motions  of  Encke's  comet,  which  sometimes  passes  near  to 
that  planet.  The  mass  of  Venus  is  determined  by  the  disturbance 
which  it  causes  in  the  orbit  of  the  earth ;  and  the  mass  of  Mars  is 
determined  in  the  same  manner. 


QUANTITY   OF   MATTER  IN  THE  SUN  AND  PLANETS.       249 

470.  How  to  determine  the  density  of  a  planet. — Having  determ- 
ined the  quantity  of  matter  in  the  sun  and  planets,  and  knowing 
also  their  volumes,  Art.  387,  we  can  compute  their  densities,  for 
these  are  proportional  to  the  masses  divided  by  the  volumes. 
Knowing  also  the  specific  gravity  of  the  earth,  Art.  49,  we  can 
compute  the  specific  gravity  of  each  member  of  the  solar  system. 
The  following  table  shows  the  mass,  density,  and  specific  gravity 
of  the  principal  members  of  our  solar  system.  The  masses  are 
according  to  Le  Verrier,  and  differ  somewhat  from  the  results  of 
the  preceding  computations. 


Mass. 

Density. 

Specific  Gravity. 

Sun     .... 

354936 

0.25 

1.37 

Mercury  .     .     . 

0.12 

2.01 

10.97 

Venus     .     .     . 

0.88 

0.97 

5.30 

Earth.     .     .     . 

1.00 

1.00 

5.46 

Mars  .... 

0.13 

0.72 

3.93 

Jupiter    .    .     . 

338.03 

0.24 

1.31 

Saturn     .     .     . 

101.06 

0.13 

0.71 

Uranus    .     .     . 

14.79 

0.15 

0.82 

Neptune  .     .     . 

24.65 

0.27 

1.47 

Upon  comparing  the  numbers  in  this  table,  we  do  not  readily 
perceive  any  law  connecting  the  density  of  a  planet  with  its  dis- 
tance from  the  sun.  The  four  outer  planets  have  a  specific  grav- 
ity differing  but  little  from  that  of  water,  while  the  other  planets 
increase  in  density  somewhat  according  to  their  proximity  to  the 
sun. 

471.  Problems. 

Prob.  1.  How  much  must  the  mass  of  the  earth  be  increased,  in 
order  that  the  moon  might  revolve  about  it  in  the  same  time  as 
at  present,  although  removed  to  twice  her  present  distance? 

Prob.  2.  How  much  must  the  mass  of  the  earth  be  increased  to 
make  the  moon,  at  her  present  distance,  revolve  once  in  two  days? 

Prob.  3.  If  the  earth's  mass  were  350,000  times  as  great  as  at 
present,  in  what  time  would  the  moon,  at  her  present  distance,  re- 
volve around  it  ? 

Prob.  4.  What  would  be  the  periodic  time  of  a  small  body  re- 
volving about  the  moon,  at  a  distance  of  5000  miles  from  the 
moon's  centre,  assuming  the  mass  of  the  moon  to  be  -g^th  of  the 
mass  of  the  earth  ? 


250 


ASTROXOMY. 


Fig.  116. 


Prob.  5.  What  would  be  the  periodic  time  of  a  satellite  revolv- 
ing about  Jupiter  close  to  the  surface  of  the  planet? 

Prob.  6.  How  much  faster  must  Jupiter  rotate  upon  his  axis  in 
order  that  a  body  on  the  equator  of  the  planet  may  lose  all  its 
gravity  ? 

PERTURBATIONS  OF  THE  PLANETS. 

472.  How  to  compute  the  disturbing  force  of  apland, — It  appears 
from  Art.  470  that  the  mass  of  the  sun  is  more  than  a  thousand 
times  greater  than  the  largest  planet,  and  more  than  a  hundred 
thousand  times  greater  than  the  smaller  planets.  Moreover,  the 
difference  between  the  mean  distances  of  the  planets  is  so  great, 
and  the  eccentricities  of  their  orbits  are  so  small,  that,  when  they 
approach  nearest  to  each  other,  the  disturbing  force  exerted  by 
any  one  upon  any  other  is  only  a  minute  fraction  of  the  attrac- 
tion of  the  sun.  Both  the  intensity  and  direction  of  the  disturb- 
ing force  caused  by  any  one  of  the  planets  may  be  computed  in 
the  same  manner  as  was  shown  in  the  case  of  the  moon,  Art.  269. 
Let  S  represent  the  sun,  P  a  planet  revolv- 
ing in  its  orbit  ABCD,  and  let  M  be  another 
planet  which,  by  its  attraction,  disturbs  the 
motion  of  P.  Take  SM  to  represent  the  force 
with  which  M  attracts  S ;  and  in  the  line  PM, 
produced  if  necessary,  take  PE  such  that 
PE  :  MS : :  MS2 :  MP2 ;  then  PE  will  repre- 
sent in  quantity  and  direction  the  force  with 
which  M  attracts  P.  Eesolve  PE  into  PF 
and  PG,  of  which  PF  is  equal  and  parallel  to 
SM.  Then,  as  in  Art.  269,  PG  represents  the 
disturbing  force  of  M  upon  P. 

The  ratio  of  the  line  PG  to  SM  may  be 
computed  by  Trigonometry  when  we  know 
the  distances  of  the  two  planets  from  the  sun, 
and  also  their  relative  situations.  The  dis- 
turbing force  of  M  upon  P  may  then  be  com- 
pared with  the  sun's  attraction  on  P  by 
means  of  the  following  proportions : 

1.  Disturbing  force :  M's  attraction  on  sun  : :  PG :  SM. 

2.  M's  att.  on  sun  :  sun's  att.  on  M  : :  M's  mass  :  sun's  mass. 

3.  Sun's  attraction  on  M  :  sun's  attraction  on  P : :  SP2 :  SM3. 


PERTURBATIONS  OF  THE  PLANETS.          251 

Compounding  these  proportions,,  we  have, 

Disturbing  force :  sun's  attraction  on  P : :  PG  x  SP2  x  M's  mass : 
SM3  x  sun's  mass. 

473.  Disturbing  force  of  Jupiter  and  Saturn. — In  order  to  show 
the  application  of  these  principles,  we  will  compute  the  disturbing 
force  of  the  two  largest  planets,  Jupiter  and  Saturn,  upon  each 
other  in  two  different  positions. 

Ex.  1.  Compare  the  disturbing  force  of  Saturn  upon  Jupiter 
with  the  sun's  attraction  upon  Jupiter  when  the  two  planets  are 
in  conjunction,  assuming  the  distances  of  Jupiter  and  Saturn  from 
the  sun  to  be  5.2028  and  9.5388,  and  the  mass  of  the  sun  to  be 
3512  times  that  of  Saturn. 

Saturn's  att.  on  Jupiter :  Saturn's  att. 
on  sun  : :  MS2  :  MP2 : :  9.53S82 :  4.3362  m 


::  4.8396:1. 

The  force  with  which  Saturn  draws  Jupiter  away  from  the  sun 
is  therefore  represented  by  3.8396 ;  or, 

Disturbing  force  :  Saturn's  attraction  on  sun  ::  3.8396  ;  1. 

Saturn's  att.  on  sun  :  sun's  attraction  on  Saturn  : :  1 :  3512. 

Sun's  att.  on  Saturn  :  sun's  att.  on  Jupiter : :  5.2022 :  9.53S2. 

By  compounding  these  proportions,  we  have, 
Disturbing  force  :  sun's  attraction  on  Jupiter  : :  3.8396  x  5.2022 : 

3512  X9.5382::  1:3075; 

that  is,  by  the  disturbing  action  of  Saturn  at  conjunction,  Jupiter's 
gravity  to  the  sun  is  diminished  by  -guVsth  part. 

Ex.  2.  Compare  the  disturbing  force  of  Saturn  upon  Jupiter 
with  the  sun's  attraction  upon  Jupiter  when  the  two  planets  are 
in  opposition  to  each  other.  Ans.  The  ratio  is  1 :  20300 ; 

that  is,  by  the  disturbing  action  of  Saturn  at  opposition,  Jupiter's  grav' 
ity  to  the  sun  is  diminished  by  -j^thyTrth  part. 

Ex.  3.  Compare  the  disturbing  force  of  Jupiter  upon  Saturn 
with  the  sun's  attraction  upon  Saturn  when  the  two  planets  are 
in  conjunction,  assuming  the  mass  of  the  sun  to  be  1050  times 
tfiat  of  Jupiter. 

Jupiter's  att,  on  Saturn  :  Jupiter's  att.  n  ijg 

on  sun  : :  MS2  :  MP2  : :  5.20282 :  4.3362  s *'     M  p 

: :  1.4397  :  1. 

Hence  the  force  with  which  Jupiter  draws  Saturn  toward  the 
sun  is  represented  by  2.4397 ;  or, 


252  ASTRONOMY. 

Disturbing  force  :  Jupiter's  attraction  on  sun  : :  2.4397  : 1. 

Jupiter's  att.  on  sun  :  sun's  attraction  on  Jupiter  : :  1 : 1050. 

Sun's  att  on  Jupiter :  sun's  att  on  Saturn  : :  9.538* :  5.2022. 

By  compounding  these  proportions,  we  have, 
Disturbing  force:  sun's  attraction 'on  Saturn  ::  2.4397x9.538*: 

1050  x  5.2022::  1:128 ; 

that  is,  by  the  disturbing  action  of  Jupiter  at  conjunction,  Saturn's 
gravity  to  the  sun  is  increased  by  i\-g\\\  part. 

Ex.  4.  Compare  the  disturbing  force  of  Jupiter  upon  Saturn 
with  the  sun's  attraction  upon  Saturn  when  the  two  planets  are 
in  opposition  to  each  other.  Ans.  The  ratio  is  1 :  357; 

that  is,  by  the  disturbing  action  of  Jupiter  at  opposition,  /Saturn's 
gravity  to  the  sun  is  diminished  by  Trsrth  part. 

474.  Periodical  inequalities  of  the  planets. — It  is  thus- seen  to  be 
possible  to  compute  the  direction  and  intensity  of  the  disturbing 
force  exerted  at  any  time  by  any  planet  upon  any  other  planet ; 
we  can  therefore  compute  how  much  each  planet  will  be  drawn 
out  of  its  elliptic  path  by  the  disturbing  action  of  the  other  plan- 
ets.    These  disturbances,  as  stated  in  Art.  280,  are  either  period- 
ical or  secular.     The  periodical  inequalities  of  the  planets  are 
generally  small.     Those  of  Mercury  can  never  exceed  a  quarter 
of  a  minute;  those  of  Venus  can  never  exceed  half  a  minute; 
those  of  the  earth  about  one  minute ;  and  those  of  Mars  about 
two  minutes.     Those  of  Jupiter  may  amount  to  20  minutes,  and 
those  of  Saturn  to  48  minutes,  while  those  of  Uranus  are  less  than 
3  minutes. 

475.  Long  inequalities. — Some  of  these  inequalities  are  very  re- 
markable for  the  length  of  their  periods,  arising  from  a  near  ap- 
proach to  commensurability  in  the  times  of  revolution.     Eight 
times  the  period  of  the  earth  is  nearly  equal  to  thirteen  times  the 
period  of  Venus ;  or  235  times  the  period  of  the  earth  is  almost 
exactly  equal  to  382  times  the  period  of  Venus,  as  was  shown 
Art.  409.     Hence  arises  in  the  motion  of  both  of  these  planet^ 
an  inequality  having  a  period  of  235  years ;  amounting,  however, 
to  only  2".9  for  Venus,  and  to  2".0  for  the  earth. 

476.  Long  inequality  of  Jupiter  and  Saturn.  —  The  long  ine- 
quality in  the  motion  of  Jupiter  and  Saturn  is  very  celebrated  in 


PERTURBATIONS  OF  THE  PLANETS.  253 

the  history  of  Astronomy.  Five  times  the  period  of  Jupiter  is 
nearly  equal  to  twice  that  of  Saturn ;  or  77  revolutions  of  Jupiter 
are  very  nearly  equal  to  31  of  Saturn,  corresponding  to  a  period 
of  913  years.  Hence  arises  in  the  motion  of  both  of  these  plan- 
ets an  inequality  having  a  period  of  over  900  years,  amounting,  at 
its  maximum,  in  the  case  of  Jupiter,  to  about  20',  and  in  the  case 
of  Saturn  to  48'. 

477.  Long  inequality  of  Uranus  and  Neptune. — There  is  a  similar 
inequality  in  the  motions  of  Uranus  and  Neptune.     The  periodic 
time  of  Neptune  is  nearly  double  that  of  Uranus;  or,  more  accu- 
rately, 25  revolutions  of  Neptune  correspond  to  49  of  Uranus. 
Hence  arises  in  the  motions  of  these  planets  an  inequality  having 
a  period  of  over  4000  years. 

478.  Secular  inequalities  of  the  planets. — The  secular  inequalities 
of  the  planets  are  generally  small,  but  in  the  lapse  of  time  become 
important  by  their  continued  accumulation.    The  nodes  of  all  the 
planetary  orbits  have  a  slow  motion  westward  on  the  ecliptic, 
amounting,  in  one  case,  to  36"  annually.     The  line  of  the  apsides 
of  their  orbits  is  also  in  continual  motion,  that  of  Mercury  moving 
eastward  5",  that  of  the  earth  12",  and  that  of  Saturn  19"  annually. 

The  disturbing  action  of  one  planet  upon  another  causes  the 
line  of  the  apsides  sometimes  to  progress,  and  at  other  times  to 
regress ;  but  in  the  case  of  most  of  the  planets  the  former  effect 
predominates. 

479.  Secular  variation  of  the  inclination. — The  inclinations  of  the 
planetary  orbits  and  their  eccentricities  are  only  subject  to  small 
periodical  variations  on  each  side  of  a  mean,  from  which  they 
never  greatly  depart.     In  no  case  (excepting  the  asteroids)  does 
the  change  of  inclination  amount  to  one  half  of  a  second  annually. 
The  inclinations  of  the  orbits  of  Jupiter  and  Saturn  are  closely 
related  to  each  other ;  and  it  has  been  computed  that  the  inclina- 
tion of  the  orbit  of  Jupiter  to  the  ecliptic  must  oscillate  between 
the  values  of  2°  2'  and  1°  17',  while  that  of  Saturn  will  oscillate 
between  the  values  of  2°  32'  and  0°  46',  requiring  for  these  changes 
a  period  of  50,000  years. 

The  inclination  of  the  earth's  equator  to  the  ecliptic  is  now  24 
minutes  less  than  it  was  twenty-one  centuries  ago,  and  is  now  de- 


254  ASTRONOMY. 

creasing  at  the  rate  of  half  a  second  annually ;  but  it  has  been 
proved  that  this  is  a  secular  inequality  of  a  long  period,  and,  after 
reaching  a  minimum,  will  return  in  the  contrary  direction,  and 
thus  oscillate  back  and  forth  about  a  mean  position.  It  has  been 
computed  that  the  obliquity  has  been  decreasing  for  20,000  years, 
and  will  continue  to  decrease  for  15,000  years  longer,  when  it  will 
be  reduced  to  22°  15' ;  after  which  the  obliquity  will  increase. 

480.  Secular  variation  of  the  eccentricity. — The  eccentricities  of 
all  the  planetary  orbits  are  continually  changing,  but  (with  the 
exception  of  the  asteroids)  this  change  in  no  case  exceeds  one 
thousandth  part  in  300  years.     In  every  instance  these  changes 
will  always  be  confined  within  moderate  limits.     Those  of  Ju- 
piter will  be  confined  within  the  limits  of  0.06  and  0.02,  while 
those  of  Saturn  will  be  confined  within  the  limits  of  0.08  and 
0.01,  the  period  in  each  case  being  35,000  years. 

The  eccentricity  of  the  earth's  orbit  is  decreasing  at  the  rate  of 
0.00004  in  a  century ;  but  this  change  will  always  be  confined 
within  the  limits  of  0.07  and  0.003.  The  earth's  orbit  can  there- 
fore never  become  an  exact  circle.  Le  Verrier  has  computed  that 
the  eccentricity  will  continue  to  diminish  for  24,000  years,  when 
its  value  will  be  .0033.  It  will  then  begin  to  increase,  and  at  the 
end  of  another  period  of  40,000  years  its  value  will  be  .02,  after 
which  it  will  again  slowly  decrease. 

This  diminution  in  the  eccentricity  of  the  earth's  orbit  causes 
an  acceleration  in  the  mean  motion  of  the  moon  amounting  to 
10"  in  a  century,  Art.  281.  This  acceleration  will  continue  as 
long  as  the  earth's  orbit  is  approaching  the  circular  form ;  but 
when  the  eccentricity  of  the  earth's  orbit  begins  to  increase,  the 
acceleration  of  the  moon's  mean  motion  will  be  converted  into  a 
retardation. 

481.  Secular  constancy  of  the  major  axes. — Thus  the  place  of  every 
planet  in  its  orbit  is  changed  by  the  action  of  the  other  planets, 
and  the  orbit  itself  is  changed  in  all  its  elements  but  two — in  the 
major  axis  of  the  orbit,  and  the  time  of  the  planet's  revolution. 
These  two  elements  of  every  planetary  orbit  remain  secure  against 
all  disturbance.     Moreover,  all  the  inequalities  in  the  planetary 
motions  are  periodical,  and  each  of  them,  after  a  certain  period 
of  time,  runs  again  through  the  same  series  of  changes.     Ev- 


PERTURBATIONS  OF  THE  PLANETS.          255 

ery  planetary  inequality  can  be  expressed  by  terms  of  the  form 
Asin.  ntt  or  Acos.  nt,  where  A  is  a  constant  coefficient,  and  n  a 
certain  multiplier  of  t,  the  time ;  so  that  nt  is  an  arc  of  a  circle 
which  increases  proportionally  to  the  time.  Now,  although  nt  is 
thus  capable  of  indefinite  increase,  yet,  since  sin.  nt  can  never  ex- 
ceed the  radius,  or  unity,  the  inequality  can  never  exceed  A.  Ac- 
cordingly, the  value  of  the  term  Asin.  nt  first  increases  from  0  to 
A,  and  then  decreases  from  A  to  0 ;  after  which  it  becomes  nega- 
tive, and  extends  to  —A,  and  from  thence  to  0  again,  the  period 
of  all  these  changes  depending  on  rz,  the  multiplier  of  t.  If  the 
value  of  any  of  the  planetary  inequalities  contained  a  term  of  the 
form  Ant,  or  Atang.  nt,  the  inequality  so  expressed  would  increase 
without  limit.  Lagrange,  Laplace,  and  Poisson,  in  demonstrating 
that  no  such  terms  as  these  last  can  enter  into  the  expression  of 
the  disturbances  of  the  planets,  made  known  one  of  the  most  im- 
portant truths  in  physical  astronomy.  They  proved  that  the 
planetary  system  is  stable;  that  the  planets  will  neither  recede  in- 
definitely from  the  sun,  nor  fall  into  it,  but  continue  to  revolve 
forever  in  orbits  of  very  nearly  the  same  dimensions  as  at  pres- 
ent, unless  there  is  introduced  the  action  of  some  external  force. 

482.  Why  the  solar  system  is  stable. — This  accurate  compensation 
of  the  inequalities  of  the  planetary  motions  depends  on  certain 
conditions  belonging  to  the  original  constitution  of  the  solar  sys- 
tem. 

1st.  It  is  essential  that  the  mass  of  the  central  body  should  be 
much  greater  than  that  of  any  of  the  planets.  If  the  mass  of  the 
sun  were  no  greater  than  that  of  Jupiter,  then  the  disturbing  ac- 
tion of  Jupiter  upon  the  nearer  planets  would  be  sufficient  en- 
tirely to  change  the  form  of  their  orbits ;  but  at  present  the  dis- 
turbing action  of  Jupiter  upon  any  one  of  the  planets  is  small 
when  compared  with  the  attraction  of  the  sun  upon  the  same 
planet. 

2d.  It  is  essential  that  the  distances  between  the  planets,  espe- 
cially of  the  larger  ones,  should  be  considerable  when  compared 
with  their  distances  from  the  sun,  otherwise,  when  in  conjunction, 
their  disturbing  action  upon  each  other  would  be  so  great  as  en- 
tirely to  change  the  form  of  their  orbits.  Hence,  also, 

3d.  It  is  essential  that  the  orbits  of  the  planets,  especially  of  the 
larger  ones,  should  have  but  little  eccentricity.  If  the  orbit  of 


256  ASTRONOMY. 

Jupiter  were  as  eccentric  as  that  of  many  of  the  comets,  he  might 
at  times  approach  so  near  to  the  earth  as  entirely  to  change  the 
form  of  our  orbit  This  principle  is  well  illustrated  by  the  case 
of  the  comet  of  1770.  See  Art  521. 

4th.  It  is  essential  that  the  planets  should  revolve  around  the 
sun  in  planes  but  little  inclined  to  each  other.  If  Jupiter's  orbit 
had  great  inclination  to  the  ecliptic,  he  would  tend  continually  to 
draw  the  earth  out  of  its  present  plane  of  motion,  while  the  axis 
of  the  earth  would  retain  a  fixed  position  in  space ;  that  is,  the 
obliquity  of  the  ecliptic  might  change  very  greatly,  and  this  would 
involve  a  change  of  seasons  which  might  be  very  unfavorable 
both  to  animal  and  vegetable  life. 

These  four  conditions  are  essential  to  the  stability  of  any  sys- 
tem. In  the  demonstration  by  Laplace  that  the  solar  system  is 
stable,  a  fifth  condition  is  required,  namely,  that  the  planets  all 
move  about  the  sun  in  the  same  direction. 

These  conditions  do  not  necessarily  result  from  the  nature  of 
motion  or  of  gravitation,  neither  can  they  be  ascribed  to  chance, 
for  it  is  improbable  that  without  a  cause  particularly  directed  to 
that  object  there  should  be  such  a  conformity  in  the  motions  of 
so  many  bodies  scattered  over  so  vast  an  extent  of  space.  It 
seems  difficult  to  avoid  the  conclusion  that  all  this  is  the  work 
of  intelligence  and  design,  directing  the  original  constitution  of 
the  system  so  as  to  give  stability  to  the  whole. 

483.  Effect  of  commensurability  in  the  periodic  times. — If  the  pe- 
riodic times  of  the  planets  were  commensurable,  and  could  per- 
manently continue  thus,  it  would  endanger  the  stability  of  the 
solar  system.  If,  for  example,  Neptune  made  exactly  one  revolu- 
tion while  Uranus  makes  two  (as  it  does  very  nearly),  then  the 
two  planets  would  always  come  into  conjunction  in  the  same  part 
of  their  orbits ;  the  effect  which  Neptune  produces  upon  Uranus 
at  one  conjunction,  although  small,  would  be  doubled  at  the  sec- 
ond conjunction,  and  trebled  at  the  third;  and,  after  the  lapse  of 
a  large  number  of  revolutions,  the  orbit  of  Uranus  would  be  en- 
tirely changed.  But  eVen  supposing  the  periodic  times  to  have 
been  made  exactly  commensurable,  they  could  not  permanently 
continue  so,  since  any  change  produced  in  the  periodic  time  of 
the  disturbed  planet  is  necessarily  accompanied  by  a  change  in 
the  opposite  direction  in  that  of  the  disturbing  planet,  so  that  the 


COMETS.  257 

periods  would  become  incommensurable  by  the  mere  effect  of 
their  mutual  action.  So  long  as  the  periodic  times  of  the  planets 
are  incommensurable,  their  conjunctions  take  place  successive- 
ly upon  different  parts  of  the  orbit,  and  their  effects  compensate 
each  other  by  mutual  opposition.  When  there  is  a  near  approach 
to  commensurability  (as  in  the  case  already  cited),  it  requires  a 
long  period  of  years  for  these  effects  to  compensate  each  other — 
that  is,  it  gives  rise  to  an  inequality  of  a  long  period. 


CHAPTER  XVIII. 

COMETS. — COMETARY  ORBITS. — SHOOTING  STARS. 

484.  What  is  a  comet? — A  comet  is  a  nebulous  body  revolving 
in  an  orbit  about  the  sun,  sometimes  with  a  bright  nucleus  and 
tail,  but  frequently  with  neither.    The  orbits  of  all  known  comets 
are  more  eccentric  than  any  of  the  planets.     The  most  eccentric 
planetary  orbit  known  is  that  of  Aethra  (132),  one  of  the  asteroids, 
whose  eccentricity  is  0.380;  the  least  eccentric  cometary  orbit  is 
that  of  Faye's  comet,  whose  eccentricity  is  0.556.    In  consequence 
of  this  eccentricity,  and  of  the  faintness  of  their  illumination,  all 
comets,  during  a  part  of  every  revolution,  disappear  from  the  ef- 
fect of  distance. 

485.  Number  of  comets.  —  The  number  of  comets  which  have 
been  recorded  since  the  birth  of  Christ  is  over  600,  and  the  num- 
ber of  those  whose  orbits  have  been  computed  is  250.     Of  these, 
205  have  moved  either  in  parabolic  orbits,  or  in  ellipses  of  such 
eccentricity  that  they  could  not  be  distinguished  from  parabolas, 
there  are  five  whose  motions  are  best  represented  by  an  hyper- 
bola, while  about  40  have  been  computed  to  move  in  elliptic 
orbits. 

The  number  of  comets  belonging  to  the  solar  system  must 
amount  to  many  thousands.  Eighty  comets  have  been  recorded 
within  the  past  50  years ;  and  if  we  admit  that  the  proportion  of 
faint  comets  was  as  great  before  the  invention  of  the  telescope  as 
it  has  been  since,  we  must  conclude  that  more  than  4000  comets 
have  approached  the  sun  within  the  orbit  of  Mars  since  the  com 
mencement  of  the  Christian  era. 
R 


258  ASTRONOMY. 

Comets  are  usually  named  from  the  year  in  which  they  appear. 
Several  of  them,  however,  are  known  by  the  name  of  their  first 
discoverer,  or  of  some  astronomer  specially  connected  with  their 
history.  Among  the  most  celebrated  comets  are  those  of  Halley, 
Encke,  Biela,  Donati,  etc.,  which  are  specially  described  in  Arts. 
507-524. 

486.  Position  of  cometary  orbits. — The  orbits  of  comets  exhibit 
every  possible  variety  of  position.     Their  inclinations  to  the 
ecliptic  range  from  0°  to  90°,  and  their  motion  is  as  frequently 
retrograde  as  direct ;  in  other  words,  their  inclinations  range  from 
0°  to  180°.     Unlike  the  planets,  the  comets  are  seen  near  the 
poles  of  the  heavens  as  well  as  near  the  ecliptic. 

487.  Period  of  visibility. — The  duration  of  a  comet's  visibility 
varies  from  a  few  days  to  more  than  a  year,  but  it  most  usually 
happens  that  it  does  not  exceed  two  or  three  months.     Only  six 
comets  have  been  observed  so  long  as  8  months.     The  comet  of 
1825  was  observed  nearly  12  months,  and  that  of  1811  was  ob- 
served 17  months.     The  period  of  visibility  of  a  comet  depends 
on  its  intrinsic  brightness,  and  on  its  position  with  reference  to  the 
earth  and  sun. 

488.  The  coma,  nucleus,  tail,  etc. — The  most  splendid  comets 
consist  of  a  roundish,  and  more  or  less  condensed  mass  of  nebu- 
lous matter  termed  the  head,  from  which  issues,  in  a  direction  op- 
posite to  that  of  the  sun,  a  train  of  a  lighter  kind  of  nebulosity 
called  the  tail.    Within  the  head  is  sometimes  seen  a  bright  point, 
like  a  star  or  a  planet,  which  is  called  the  nucleus  of  the  comet. 
In  most  instances  the  centre  of  the  head  exhibits  nothing  more 
than  a  higher  degree  of  condensation  of  the  nebulous  matter, 
which  always  has  a  confused  appearance  in  the  telescope.     The 
nebulosity  which  surrounds  a  highly  condensed  nucleus  is  called 
the  coma.     In  most  instances  the  coma  is  less  than  100,000  miles 
in  diameter,  and  but  very  rarely  exceeds  200,000 ;  but  that  of 
the  comet  of  1811  exceeded  a  million  of  miles  in  diameter. 

489.  Dimensions  of  the  nucleus. — In  a  few  instances  the  diameter 
of  the  nucleus  has  been  computed  at  5000  miles ;  but  it  seldom 
exceeds  500  miles ;  and  the  majority  of  comets  have  no  bright 
nucleus  at  all. 


COMETS.  259 

It  is  probable  that  in  those  cases  in  which  the  diameter  of  the 
nucleus  has  been  estimated  at  5000  miles,  the  object  measured 
was  not  a  solid  body,  but  simply  nebulous  matter  in  a  very  high 
degree  of  condensation.  Thus  Donati's  comet  at  one  time  exhib- 
ited a  bright  nucleus  whose  diameter  was  computed  at  over  5000 
miles.  But  as  the  comet  approached  the  sun  and  increased  in 
brilliancy,  the  nucleus  steadily  decreased ;  and  when  the  comet 
was  nearest  to  the  sun,  the  diameter  of  the  nucleus  was  less  than 
500  miles.  The  nuclei  of  some  comets  have  exactly  the  appear- 
ance of  solid  bodies ;  but  the  true  nucleus,  apart  from  the  sur- 
rounding nebulosity,  is  probably  quite  small. 

490.  Variations  in  the  dimensions  of  comets, — The  real  dimen- 
sions of  the  nebulosities  of  comets  vary  greatly  at  different  dates 
during  their  visibility.     Many  of  them  contract  as  they  approach 
the  sun,  and  dilate  on  receding  from  the  sun.     This  has  been  re- 
peatedly observed  in  the  case  of  Encke's  comet,  and  the  same 
has  been  noticed  in  the  case  of  several  other  comets.     It  has 
been  conjectured  that  this  effect  may  result  from  the  change  of 
temperature  to  which  the  comet  is  exposed.     As  the  comet  ap- 
proaches the  sun,  the  vapor  which  composes  the  nebulous  enve- 
lope may  be  converted  by  intense  heat  into  a  transparent  and  in- 
visible elastic  fluid.     As  it  recedes  from  the  sun,  the  temperature 
decreasing,  this  vapor  is  gradually  condensed,  and  assumes  the 
form  of  a  visible  cloud ;  whence  the  visible  volume  of  the  comet 
is  increased,  while  its  real  volume  may  perhaps  be  diminished. 

491.  Changes  in  the  nebulosity  about  the  nucleus. — When  comets 
have  a  bright  nucleus  and  a  splendid  train,  the  nebulosity  about 
the  nucleus  undergoes  remarkable  changes  as  the  comet  approach- 
es the  sun.     The  nucleus  becomes  much  brighter,  and  throws  out 
a  jet  or  stream  of  luminous  matter  toward  the  sun.    Sometimes 
two,  three,  or  more  jets  are  thrown  out  at  the  same  time  in  dif- 
ferent directions.     This  emission  of  luminous  matter  sometimes 
continues,  with  occasional  interruptions,  for  several  weeks.     The 
form  and  direction  of  these  luminous  streams  undergo  singular 
and  capricious  alterations,  so  that  no  two  successive  nights  pre- 
sent the  same  appearance.    These  jets,  though  very  bright  at  their 
point  of  emanation  from  the  nucleus,  fade  away  and  become  dif- 
fuse as  they  expand  into  the  coma,  at  the  same  time  curving  back- 


260  ASTRONOMY. 

ward,  as  if  impelled  against  a  resisting  medium.  These  streams 
combined  form  the  outline  of  a  bright  parabolic  envelope  sur- 
rounding the  nucleus,  and  this  envelope  steadily  increases  in  its 
dimensions,  receding  from  the  nucleus.  After  a  few  days  a  sec- 
ond luminous  envelope  is  sometimes  formed  within  the  first,  the 
two  being  separated  by  a  band  comparatively  dark,  and  this  sec- 
ond envelope  steadily  increases  in  its  dimensions  from  day  to  day. 
A  few  days  later  a  third  envelope  is  sometimes  formed,  and  so  on 
for  a  long  series.  Donati's  comet  showed  seven  such  envelopes 
in  succession,  each  separated  from  its  neighbor  by  a  band  com- 
paratively dark,  and  each  steadily  receding  from  the  nucleus.  See 
Plate  VI.,  Fig.  3. 

These  envelopes  seem  to  be  formed  of  substances  different  from 
the  vapors  on  the  earth's  surface,  for  they  do  not  sensibly  refract 
light  They  appear  to  be  driven  off  from  the  nucleus  by  a  re- 
pulsive force  on  the  side  next  the  sun,  somewhat  as  light  particles 
are  thrown  off  by  electric  repulsion  from  an  excited  conductor; 
and  the  dark  bands  separating  the  successive  envelopes  seem  to 
result  from  a  periodical  cessation  or  diminished  activity  of  this 
repulsive  force. 

492.  The  tail. — The  tail  of  a  comet  is  but  the  prolongation  of 
the  nebulous  envelope  surrounding  the  nucleus.  Each  particle 
of  matter,  as  it  issues  from  the  nucleus  on  the  side  next  to  the  sun, 
gradually  changes  its  direction  by  a  curved  path,  until  its  motion 
is  almost  exactly  away  from  the  sun.  The  brightness  and  extent 
of  the  train  increase  with  the  brightness  and  magnitude  of  the  en- 
velopes, the  tail  appearing  to  consist  exclusively  of  the  matter  of 
the  envelopes,  driven  off  by  a  powerful  repulsive  force  emanating 
from  the  sun.  On  the  side  of  the  nucleus  opposite  to  the  sun 
there  is  no  appearance  of  luminous  streams,  and  hence  results  a 
dark  stripe  in  the  middle  of  the  tail,  dividing  it  longitudinally 
into  two  distinct  parts.  This  stripe  was  formerly  supposed  to  be 
the  shadow  of  the  head  of  the  comet ;  but  the  dark  stripe  still 
exists  even  when  the  tail  is  turned  obliquely  to  the  sun.  The 
tail  is  probably  a  hollow  envelope ;  and  when  we  look  at  the 
edges,  the  visual  ray  traverses  a  greater  quantity  of  nebulous  par- 
ticles than  when  we  look  at  the  central  line,  which  circumstance 
would  cause  the  central  line  to  appear  less  bright  than  the  sides. 


COMETS.  261 

493.  Rapidity  of  formation  of  the  tail. — When  comets  make  their 
first  appearance  they  generally  have  little  or  no  tail ;  but,  by  de- 
grees, the  nebulous  envelope  is  formed,  the  tail  soon  appears,  which 
increases  in  length  and  brightness  as  the  comet  approaches  peri- 
helion.    When  the  comet  is  nearest  the  sun,  the  tail  sometimes 
increases  with  immense  rapidity.     The  tail  of  Donati's  comet  in 
1858  increased  in  length  at  the  rate  of  two  millions  of  miles  per 
day ;  that  of  the  great  comet  of  1811  increased  at  the  rate  of  nine 
millions  of  miles  per  day ;  while  that  of  the  great  comet  of  1843, 
soon  after  passing  perihelion,  increased  at  the  rate  of  35  millions 
of  miles  per  day. 

494.  Dimensions  of  the  tail. — The  tails  of  comets  frequently  have 
an  immense  length.     That  of  1843  attained  a  length  of  120  mil- 
lions of  miles ;  that  of  1811  had  a  length  of  over  100  millions  of 
miles,  and  a  breadth  of  about  15  millions;  and  there  have  been 
four  other  comets  whose  tails  attained  a  length  of  50  millions  of 
miles. 

The  apparent  length  of  the  tail  depends  not  merely  upon  its 
absolute  length,  but  upon  the  direction  of  its  axis,  and  its  distance 
from  the  earth.  There  are  six  comets  on  record  whose  tails  sub- 
tended an  angle  of  90°  and  upward — that  is,  whose  length  would 
reach  from  the  horizon  to  the  zenith  ;  and  there  are  about  a  dozen 
more  whose  tails  subtended  an  angle  of  at  least  45°. 

The  tail  usually  attains  its  greatest  length  and  splendor  a  few 
days  after  the  comet  passes  its  perihelion ;  and  as  the  comet  re- 
cedes farther  from  the  sun,  the  tail  fades  gradually  away,  being 
apparently  dissipated  in  space. 

495.  Position  of  the  axis  of  the  tail — The  axis  of  the  tail,  CG, 
Fig.  119,  is  not  a  straight  line,  and,  except  near  the  nucleus,  is  not 
directed  exactly  from  the  sun,  but  alwaj^s  makes  an  angle  with  a 
radius  vector,  SO.     This  angle  generally  amounts  to  10°  or  20°, 
and  sometimes  even  more,  the  tail  always  inclining  from  the  re- 
gion toward  which  the  comet  proceeds.     If  the  tail  were  formed 
by  a  repulsive  force  emanating  from  the  sun,  which  carried  parti- 
cles instantly  from  the  comet's  head  to  the  extremity  of  the  tail, 
then  the  axis  of  the  tail  ought  to  be  turned  exactly  from  the  sun. 
But,  in  fact,  the  particles  at  the  extremity  of  the  tail,  as  G,  are 
those  which  were  emitted  from  the  nucleus  several  days  previ- 


262 


ASTRONOMY. 


ous,  perhaps  20  days,  when  the  head  of  the  comet  was  at  A,  and, 
in  consequence  of  their  inertia,  they  retain  the  motion  in  the  di- 
rection of  the  orbit  which  the  nucleus  had  at  the  time  they  part- 
ed from  it  The  particles  near  the  middle  of  the  tail,  as  H,  are 
those  which  left  the  nucleus  later  than  the  preceding,  perhaps  10 
days,  when  the  head  of  the  comet  was  at  B,  and  they  retain  the 
motion  in  the  direction  of  the  orbit  which  the  nucleus  had  at  the 
time  when  they  parted  from  it. 

496.  Probable  mode  of  formation  of  comet's  tails. — In  order  to  ex- 
plain the  phenomena  of  comet's  tails,  it  seems  necessary  to  admit 
the  existence  of  a  repulsive  force  by  which  certain  particles  of  a 
comet  are  driven  off  from  the  nucleus,  and  that  these  particles  are 
then  acted  upon  by  a  more  powerful  repulsive  force  emanating 
from  the  sun. 


Fig.  119. 


Let  S  represent  the  position  of  the  sun,  and  ABC  a  portion  of 
a  comet's  orbit,  the  comet  moving  in  the  direction  of  the  arrows. 
Suppose,  when  the  nucleus  is  at  A,  a  particle  of  matter  is  expelled 
from  the  head  of  the  comet  in  the  direction  SAD.  This  particle 
will  still  retain  the  motion  which  it  had  in  common  with  the  nu- 
cleus, and  this  motion  would  carry  the  particle  over  the  line  DG 
while  the  head  is  moving  from  A  to  C.  When  the  nucleus  reach- 
es B,  suppose  another  particle  to  be  driven  off  in  the  direction 
SBE.  This  particle  will  also  retain  the  motion  which  it  had  in 
common  with  the  nucleus,  and  which  would  carry  it  over  EH 
while  the  head  is  moving  from  B  to  C.  Thus,  when  the  nucleus 
has  reached  the  point  C,  the  particles  which  were  expelled  from 
the  head  during  the  time  of  its  motion  from  A  to  C  will  all  be 


COMETS.  263 

situated  upon  the  line  CHG.  It  is  evident  that  this  line  will  be 
a  curve,  tangent  at  C  to  the  radius  vector  SO  produced,  and  al- 
ways curving /rcw  the  region  toward  which  the  comet  proceeds. 

497.  Comets  with  several  tails. — A  transverse  section  of  the  tail 
of  a  comet  is  not  generally  a  circle,  but  an  oval  curve,  more  or 
less  elongated.     In  the  case  of  Donati's  comet,  the  greatest  diam- 
eter of  this  oval  was  about  four  times  the  least,  and  in  the  comet 
of  1744  the  ratio  was  probably  still  greater.     The  longest  diam- 
eter of  the  transverse  section  coincides  nearly  with  the  plane  of 
the  orbit;  in  other  words,  the  tail  of  a  comet  spreads  out  like  a 
fan,  so  that  its  breadth,  measured  in  the  direction  of  the  plane  of 
the  orbit,  is  greater  than  its  breadth  measured  in  a  transverse  di- 
rection. 

In  order  to  explain  this  phenomenon,  it  seems  necessary  to  ad- 
mit that  the  repulsive  force  of  the  sun  is  not  the  same  upon  all 
the  particles  which  form  the  tail  of  the  comet  Those  particles 
upon  which  the  repulsive  force  of  the  sun  is  very  great  will  form 
a  tail  which  is  turned  almost  exactly  from  the  sun  ;  but  those 
particles  upon  which  the  repulsive  force  of  the  sun  is  small  will 
form  a  tail  which  falls  very  much  behind  the  direction  of  a  radius 
vector.  If,  then,  the  head  of  the  comet  consists  of  particles  which 
are  unequally  acted  upon  by  the  sun,  the  comet  may  have  several 
tails,  or  perhaps  an  indefinite  number,  whose  axes  occupy  some- 
what different  positions,  but  'all  are  situated  in  the  plane  of  the 
orbit.  This  theory  will  enable  us  to  explain  the  striped  appear- 
ance of  the  tail  of  Donati's  comet  (see  Plate  VI.,  Fig.  2),  as  well  as 
the  faint  streamers  which  extended  in  a  direction  very  nearly  op- 
posite to  the  sun.  It  also  explains  the  very  remarkable  appear- 
ance of  the  comet  of  1744,  which  is  commonly  said  to  have  had 
six  tails.  When  near  perihelion,  the  head  of  the  comet  being  be- 
low the  horizon,  the  tail  was  seen  to  extend  above  the  horizon, 
as  represented  in  Plate  V.,  Fig.  5. 

498.  Telescopic  comets. — Most  comets  are  not  attended  by  tails. 
Telescopic  comets  seldom  exhibit  this  appendage.     They,  how- 
ever, generally  become  elongated  as  they  approach  the  sun,  and 
the  point  of  greatest  brightness  does  not  occupy  the  centre  of  the 
nebulosity. 

In  many  cases,  the  absence  of  a  tail  is  probably  owing  to  the 


264  ASTRONOMY. 

smallness  of  the  comet,  and  the  consequent  faintness  of  its  light; 
so  that,  although  a  tail  is  really  formed,  it  entirely  escapes  obser- 
vation. 

In  other  cases,  it  seems  probable  that,  by  frequent  approaches 
to  the  sun,  the  comet  has  lost  all  of  that  class  of  particles  which 
are  repelled  by  the  sun,  and  which  contribute  to  form  a  tail ;  and 
such  comets  exhibit  only  a  slight  elongation  as  they  approach  the 


499.  Quantity  of  matter  in  comets. — The  quantity  of  matter  in 
comets  is  exceedingly  small.     Comets  have  been  known  to  pass 
near  to  some  of  the  planets  and  their  satellites,  and  to  have  had 
their  own  motions  much  disturbed  by  the  consequent  attractions, 
•without  producing  any  sensible  disturbance  in  the  motion  of  the 
planets  or  their  satellites.    Since  the  quantity  of  matter  in  comets 
is  inappreciable  in  comparison  with  the  satellites,  while  their  vol- 
umes are  enormously  large,  the  density  of  the  comet's  nebulosity 
must  be  incalculably  small. 

The  transparency  of  the  nebulosities  of  comets  is  still  more  re- 
markable. Stars  of  the  smallest  magnitude  have  been  repeated- 
ly seen  through  comets  of  from  50,000  to  100,000  miles  in  di- 
ameter, and,  in  the  majority  of  cases,  not  the  least  perceptible 
diminution  of  the  star's  brightness  could  be  detected. 

500.  Do  comets  exhibit  phases? — Comets  exhibit  no  phases  like 
those  presented  by  the  moon,  and  which  might  be  expected  from 
a  solid  nucleus  shining  by  reflected  light     Some  have  therefore 
doubted  whether  comets  shine  simply  by  the  borrowed  light  of 
the  sun.     The  following  consideration  proves  that  their  light,  at 
least  for  the  most  part,  depends  on  their  distance  from  the  sun. 
A  self-luminous  surface  appears  of  the  same  brilliancy  at  all  dis- 
tances as  long  as  it  subtends  a  sensible  angle.     Thus  the  surface 
of  the  sun,  as  seen  from  Uranus,  must  appear  as  bright  as  it  does 
to  us,  only  subtending  a  smaller  angle.     If,  then,  a  comet  shines 
by  its  own  light,  it  should  retain  its  brilliancy  as  long  as  its  di- 
ameter has  a  sensible  magnitude.     Such,  however,  is  not  the  case. 
Comets  gradually  become  dim  as  their  distance  increases ;  and 
they  vanish  simply  from  loss  of  light,  while  they  still  retain  a 
sensible  diameter. 


COMETS.  £65 

501.  Cometary  orbits. — It  was  first  demonstrated  by  Newton  that 
a  body  which  revolves  under  the  influence  of  a  central  force  like 
gravitation,  whose  intensity  decreases  as  the  square  of  the  dis- 
tance increases,  must  move  in  one  of  the  conic  sections ;  that  is, 
either  a  parabola,  an  ellipse,  or  an  hyperbola.    Several  comets  are 
known  to  move  in  ellipses  of  considerable  eccentricity;  the  orbits 
of  most  comets  can  not  be  distinguished  from  parabolas ;  while  a 
few  have  been  thought  to  move  in  hyperbolas.     Since  the  para- 
bola and  hyperbola  consist  of  two  indefinite  branches  which  di- 
verge from  each  other,  a  body  moving  in  either  of  these  curves 
would  not  complete  a  revolution  about  the  sun.    It  would  enter 
the  solar  system  from  an  indefinite  distance,  and,  passing  through 
its  perihelion,  issue  in  a  different  direction,  moving  off  to  an  in- 
definite distance,  never  to  return.     Hence  bodies  moving  in  para- 
bolas and  hyperbolas  are  not  periodic ;  but  comets  moving  in  el- 
liptic orbits  must  make  successive  revolutions  like  the  planets. 

It  is,  however,  probable  that  the  orbits  which  are  treated  as 
parabolic  are  in  fact  very  long  ellipses,  which  differ  but  little 
from  parabolas  in  that  portion  described  by  the  comet  while  it  is 
visible. 

502.  How  to  deduce  the  orbit  of  a  comet  from  the  observations. — 
The  methods  explained  in  Chapter  XIV.  for  determining  the  or- 
bits of  the  planets  are  generally  quite  inapplicable  to  the  comets, 
because  these  methods  require  observations  to  be  made  in  particu- 
lar portions  of  the  orbit,  and  often  involve  an  interval  of  several 
years  between  the  observations;  but  comets  generally  continue 
in  sight  only  a  few  weeks,  and  from  these  few  observations  it  is 
required  to  deduce  the  form  and  position  of  the  orbit.     Sir  Isaac 
Newton  first  pointed  out  the  method  of  computing  the  orbit  of  a 
comet  from  three  observations  of  its  position.     This  method  was 
published  in  the  Principia  in  1687,  accompanied  by  a  computa- 
tion of  the  orbit  of  the  remarkable  comet  of  1680.     This  method 
has  since  been  much  simplified,  and  tables  have  been  prepared  by 
which  the  computations  are  greatly  facilitated. 

In  order  to  determine  the  orbit  of  a  comet,  we  must  therefore 
know  its  direction  in  the  heavens  on  three  different  days.  These 
observations  may  be  embraced  within  an  interval  of  48  hours ; 
but  the  longer  the  interval,  the  more  reliable  will  be  the  orbit  de- 
duced from  the  observations.  The  computations  are  also  much 


266  ASTRONOMY. 

simplified  when  the  interval  from  the  first  to  the  second  observe 
tion  is  exactly  equal  to  the  interval  from  the  second  to  the  third, 
although  this  restriction  is  by  no  means  a  necessary  one.  We 
will  suppose,  then,  that  we  have  three  observations  of  a  comet's 
place  as  seen  from  the  earth.  These  places  are  usually  denoted 
by  right  ascensions  and  declinations.  We  begin  with  converting 
these  places  into  longitudes  and  latitudes,  which  have  reference  to 
the  ecliptic,  because  we  find  it  most  convenient  to  refer  the  com* 
et's  motion  to  the  plane  of  the  earth's  orbit 

We  now  take  from  the  Nautical  Almanac  the  longitudes  of  the 
sun  for  the  same  three  instants  of  observation.  These  longitudes, 
increased  by  180°,  represent  the  longitudes  of  the  earth  as  seen 
from  the  sun.  We  then  construct  a  diagram  representing  tho 
earth's  orbit,  and  set  off  upon  it  the  places  of  the  earth  at  the 
three  given  dates.  From  each  of  these  points  we  draw  a  line  rep- 
resenting the  direction  in  which  the  comet  was  seen  at  the  cor- 
responding date.  From  these  data  it  is  required  to  deduce  the 


503.  Principles  assumed  in  computing  the  orbit. — In  computing 
the  orbit,  we  assume  certain  laws  which  have  been  verified  in  the 
case  of  all  the  known  members  of  the  solar  system.  These  laws 
are, 

1st.  The  plane  of  the  orbit  of  the  comet  must  pass  through  the 
sun. 

2d.  The  path  described  by  the  comet  must  be  a  conic  section, 
of  which  the  sun  occupies  one  of  the  foci;  and  since  we  are  sure 
that  its  orbit  is  quite  eccentric,  we  know  that  it  can  not  differ 
much  from  a  parabola.  We  therefore  assume,  in  the  first  case, 
that  the  orbit  is  a  parabola. 

3d.  The  motion  of  every  heavenly  body  in  its  orbit  about  the 
sun  is  such  that  the  areas  described  by  the  radius  vector  are  pro- 
portional to  the  times  in  which  they  are  described.  Hence  any 
area  divided  by  the  time  gives  a  quotient  which  is  a  constant 
quantity  for  the  same  body,  whether  it  be  a  planet  or  a  comet. 

4th.  For  different  bodies  revolving  about  the  sun,  the  squares 
of  the  quotients  thus  obtained  are  proportional  to  the  parameters 
of  the  orbits. 

In  applying  these  principles  to  determine  the  orbit  of  a  comet, 
we  may  first  assume  any  position  for  the  plane  of  the  orbit.  If 


COMETS.  267 

this  assumption  does  not  violate  any  of  the  preceding  principles, 
we  may  be  sure  that  we  have  found  the  true  plane  of  the  orbit. 
Otherwise  we  must  vary  the  position  of  this  plane  until  we  have 
found  one  which  does  not  conflict  with  either  of  these  principles. 
We  next  compute  the  comet's  places  from  day  to  day  in  the  sup- 
posed orbit,  and  compare  the  observed  places  with  the  computed 
places.  These  differences  must  not  exceed  the  unavoidable  errors 
of  observation.  If  they  do,  we  must  vary  the  assumed  orbit  until 
the  observed  places  agree  with  the  computed  places,  within  the 
limits  of  those  errors  to  which  such  observations  are  always  lia- 
ble. We  are  thus  able  to  decide  whether  the  orbit  is  truly  a  pa- 
rabola, an  ellipse,  or  an  hyperbola,  independently  of  any  error  in 
our  first  hypothesis. 

604.  Method  illustrated  by  a  diagram. — Let  S  represent  the  posi- 
Fj   12Q  tion  of  the  sun,  and  A,  B,  C 

the  positions  of  the  earth 
in  its  orbit  at  the  dates  of 
the  three  observations,  the 
lengths  and  position  of  the 
lines  SA,  SB,  SC  being  giv- 
en in  the  Nautical  Almanac. 
From  A  draw  a  line,  AD, 
to  represent  the  direction  in 
which  the  comet  was  seen 
\B  at  the  first  observation ;  and 
from  B  and  C,  in  like  man- 
ner, draw  lines  to  represent 
the  directions  of  the  comet 
at  the  second  and  third  observations.  We  know  that  at  the  date 
of  the  first  observation  the  comet  was  somewhere  on  the  line  AD, 
but  we  do  not  know  at  what  point  of  this  line.  If  we  assume 
that  the  comet  was  at  G,  then,  by  Principle  1st  of  Art.  503,  its 
places  at  the  other  two  dates  will  not  only  be  on  the  lines  BE, 
OF,  but  in  a  plane  passing  through  S  and  G.  Also,  by  Principle 
3d,  Art.  503,  the  parabolic  sectors  SHG,  SHK,  must  be  equal  to 
each  other.  If  the  interval  between  the  observations  is  only  a 
few  days,  the  parabolic  sectors  will  differ  but  little  from  the  plane 
triangles  SHG,  SHK,  which  must  therefore  be  nearly  equal. 
These  conditions  alone  will  generally  enable  us  to  determine  the 


268  ASTRONOMY. 

position  of  the  comet's  orbit  very  nearly.  If  the  parabolic  sec- 
tors are  found  to  be  exactly  equal,  and  if  the  magnitude  of  these 
sectors  is  such  as  is  given  by  Principle  4th,  Art.  503,  we  may  be 
sure  that  we  have  discovered  the  true  orbit.  By  this  method  we 
may,  in  less  than  an  hour,  deduce  the  approximate  orbit  of  any 
comet  from  three  observations,  embracing  an  interval  of  only  a 
few  days. 

505.  Mode  of  computing  the  orbit. — The  actual  computation  of 
the  elements  of  a  cometary  orbit  is  founded  upon  the  same  prin- 
ciples.    The  geometrical  relations  here  stated  are  represented  by 
equations,  and  these  equations  are  solved  by  successive  approxi- 
mations.    Tables  have  been  prepared  which  greatly  facilitate  the 
computation,  so  that  the  approximate  elements  of  a  cometary  or- 
bit can  be  obtained  by  the  labor  of  a  few  hours.     The  most  accu- 
rate possible  determination  of  the  orbit  is  only  obtained  by  a  care- 
ful comparison  of  all  the  observations  made  during  the  entire  pe- 
riod of  visibility,  and  this  may  involve  a  labor  of  several  weeks. 

The  method  here  indicated  is  applicable  to  the  determination 
of  the  orbit  of  a  planet  as  well  as  that  of  a  comet.  The  motions 
of  planets  and  comets  are  governed  by  exactly  the  same  laws; 
and  there  is  no  essential  difference  in  the  mode  of  computing  the 
orbits,  except  that  we  generally  assume  that  a  comet  moves  in  a 
parabola,  which  is  a  conic  section  whose  eccentricity  is  known  ; 
in  other  words,  a  cometary  orbit  involves  one  less  unknown  quan- 
tity to  be  determined  than  a  planetary  orbit.  The  methods  of 
determining  the  planetary  elements  explained  in  Chapter  XIV. 
are  applicable  to  the  brighter  planets ;  but  when  a  new  asteroid 
is  discovered,  we  are  required  to  deduce  an  orbit  at  once  from  ob- 
servations of  a  few  days,  and  this  is  accomplished  by  the  method 
here  indicated. 

506.  How  a  comet  w  known  to  be  periodical. — Since  comets  are 
only  seen  in  that  part  of  their  orbit  which  is  nearest  to  the  sun, 
and  since  an  ellipse,  a  parabola,  and  an  hyperbola,  for  a  consider- 
able distance  from  perihelion,  do  not  depart  very  widely  from 
each  other,  it  is  difficult  to  determine  in  which  of  these  curves  a 
comet  actually  moves ;  but  if  a  comet  have  an  elliptic  orbit,  it 
must  return  to  perihelion  after  completing  its  revolution.     If, 
then,  we  find  that  two  comets,  visible  in  different  years,  moved  in 


COMETS.  269 

the  same  path,  the  presumption  is  that  they  were  the  same  body, 
reappearing  after  having  completed  its  circuit  in  an  elliptic  orbit ; 
and  if  the  comet  has  been  observed  at  several  returns,  this  evi- 
dence may  amount  to  absolute  demonstration. 

The  shortest  periodic  time  of  any  comet  at  present  known  is 
3£  years  j  the  longest  period  which  has  been  positively  verified 
by  the  return  of  the  comet  is  75  years ;  but  there  are  several  com- 
ets whose  period  has  been  computed  to  exceed  a  century ;  and 
the  periods  of  some  comets  probably  amount  to  many  centuries. 
Indeed,  if  a  comet  moves  in  a  parabolic  orbit,  its  periodic  time 
must  be  infinitely  long ;  that  is,  it  could  never  complete  a  revolu- 
tion about  the  sun. 

There  are  eight  comets  whose  periods  have  been  well  establish- 
ed, viz.,  Halley's,  Encke's,  Biela's,  Faye's,  Brorsen's,  D'Arrest's, 
Winnecke's,  and  Tuttle's. 

Halley's  Comet. 

507.  Soon  after  the  publication  of  Newton's  Principia,  Ilalley, 
an  eminent  English  astronomer,  computed  from  recorded  obser- 
vations the  elements  of  a  number  of  comets  according  to  the  meth- 
od furnished  by  Newton.    These  elements  were  published  in  1705. 
On  comparing  these  orbits,  he  found  that  a  comet,  which  had  been 
observed  by  himself  and  others  in  1682,  followed  a  path  which 
coincided  very  nearly  with  those  of  cornets  which  had  been  ob- 
served in  1607  and  1531.     This  led  him  to  suppose  that,  instead 
of  three  different  comets,  it  might  be  the  same  comet  revolving  in 
an  orbit  whose  period  was  75  or  76  years.     lie  accordingly  pre- 
dicted the  reappearance  of  this  body  in  1758-9.     He  observed, 
however,  that  as,  in  the  interval  between  1607  and  1682,  the  comet 
passed  near  Jupiter,  its  velocity  must  have  been  augmented,  and, 
consequently,  its  period  shortened  by  the  action  of  that  planet, 
the  comet  ought  not  to  be  expected  to  appear  until  the  end  of 
1758,  or  the  beginning  of  1759. 

508.  Predicted  return  of  Halley's  comet. — As  the  time  approached 
for  the  fulfillment  of  this  prediction,  two  French  astronomers, 
Clairaut  and  Lalande,  undertook  to  compute  the  disturbing  effect 
of  the  planets  upon  the  comet,  and  thus  determine  its  exact  path, 
with  the  time  of  its  return  to  perihelion.    The  result  of  these  com- 
putations was  to  fix  upon  April  13, 1759,  as  the  time  of  perihelion 


270  ASTRONOMY. 

passage.  Clairaut,  however,  stated  that,  on  account  of  the  small 
quantities  unavoidably  neglected  in  his  computations,  the  time 
thus  assigned  might  vary  from  the  truth  to  the  extent  of  a  month. 
The  comet  passed  its  perihelion  on  the  12th  of  March,  just  one 
month  before  the  time  announced  by  Clairaut;  and  Laplace  has 
shown  that  if  Clairaut  had  used  in  his  calculations  the  mass  of 
Saturn  as  at  present  received,  his  prediction  would  have  been  in 
error  only  13  days. 

Before  the  comet's  next  return  in  1835,  its  path  was  computed 
by  several  astronomers,  and  the  most  complete  computations  fixed 
the  time  of  perihelion  at  November  14, 1835.  It  actually  passed 
its  perihelion  on  the  16th  of  November. 

The  mean  distance  of  this  comet  from  the  sun  is  about  18  times 
that  of  the  earth,  or  a  little  less  than  the  mean  distance  of  Ura- 
nus ;  but,  on  account  of  the  eccentricity  of  its  orbit,  its  distance 
from  the  sun  at  aphelion  is  considerably  greater  than  that  of  Nep- 
tune. 

509.  Physical  peculiarities  of  Halley's  comet. — At  its  return  in 
1835,  Halley's  comet  exhibited  physical  changes  remarkable  for 
their  magnitude  and  rapidity.  The  tail  began  to  be  formed  about 
a  month  before  the  perihelion  passage,  and  commenced  with  an 
emanation  of  nebulous  matter  from  that  part  of  the  comet  which 
was  turned  toward  the  sun.  This  emanation  resembled  a  brush 
of  electric  light  from  a  pointed  wire  in  a  dark  room.  As  this 
matter  receded  from  the  head,  it  seemed  to  encounter  a  resistance 
from  the  sun,  by  which  it  was  driven  back,  and  carried  out  to  vast 
distances  behind  the  nucleus,  forming  the  tail.  This  emanation 
took  place  only  at  intervals ;  and  sometimes  the  nebulous  matter 
thus  emitted  presented  the  appearance  of  a  second  tail  turned  to- 
ward the  sun.  At  one  time  two,  and  at  another  time  three  nebu- 
lous streams  were  observed  to  issue  in  diverging  directions.  See 
Plate  V.,  Fig.  2.  These  directions  were  continually  varying,  as 
well  as  the  comparative  brightness  of  the  emanations.  Sometimes 
they  assumed  the  form  of  a  swallow-tail.  These  jets,  though  very 
bright  at  their  point  of  emanation  from  the  nucleus,  faded  rapidly 
away,  and  became  diffused  as  they  expanded  into  the  coma. 

The  tail  seemed  to  be  formed  entirely  of  matter  thus  emitted 
from  the  head  and  repelled  by  the  sun.  The  velocity  with  which 
this  matter  was  driven  from  the  sun  was  enormous,  amounting  to 


COMETS.  271 

not  less  than  two  millions  of  miles  per  day.  On  account  of  the 
feeble  attraction  of  the  nucleus,  the  matter  thus  repelled  from  the 
head  must  mostly  escape,  and  be  lost  in  space,  never  to  reunite 
with  the  comet.  Hence  it  seems  inevitable  that  at  each  approach 
to  the  sun  the  comet  must  lose  some  of  those  particles  on  which 
the  production  of  the  tail  depends,  so  that  at  each  return  the  di- 
mensions of  this  appendage  must  become  smaller  and  smaller. 
Upon  comparing  the  different  descriptions  of  this  comet  at  its  suc- 
cessive returns  to  perihelion,  it  has  been  concluded  that  it  is  now 
much  smaller  than  it  was  in  1305.  But  the  appearances  in  1835 
did  not  indicate  any  material  diminution  since  1759,  so  that  we 
must  conclude  that,  if  this  comet  is  actually  wasting  away,  the 
process  is  a  very  gradual  one. 

EnckJs  Comet. 

510.  The  periodicity  of  this  comet  was  discovered  in  1819  by 
Professor  Encke,  of  Berlin,  who  identified  the  comet  of  that  year 
with  those  that  had  been  observed  in  1786, 1795,  and  1805,  and 
which  had  been  supposed  to  be  different  comets.    He  found  its 
period  to  be  only  about  1207  days,  or  3£  years,  and  he  predicted 
its  return  in  1822.     This  prediction  was  verified,  and  the  comet 
has  been  observed  at  every  subsequent  return  to  the  sun,  making 
13  apparitions  since  1819,  and  17  returns  for  which  we  have  ac- 
curate observations. 

At  perihelion  this  cornet  passes  within  the  orbit  of  Mercury, 
while  at  aphelion  its  distance  from  the  sun  is  fths  that  of  Jupiter. 

511.  Indications  of  a  resisting  medium.— By  comparing  observa- 
tions made  at  the  successive  returns  of  this  comet,  it  is  found  that 
the  periodic  time,  and,  consequently,  the  mean  distance  from  the 
sun,  is  subject  to  a  slow  but  regular  decrease,  amounting  to  about 
a  day  in  eight  revolutions.     It  also  appears  that  this  diminution 
is  not  produced  by  the  disturbing  action  of  the  planets.     In  order 
to  explain  the  observed  fact,  Encke  assumes  that  the  interplan- 
etary spaces  are  pervaded  by  an  extremely  rare  medium,  which 
causes  no  sensible  obstruction  to  the  motions  of  dense  bodies  like 
the  planets,  but  which  sensibly  resists  the  motion  of  a  mere  mass 
of  vapor  like  a  comet.     The  effect  of  such  a  resistance  would  be 
a  diminution  of  the  comet's  orbital  velocity,  in  consequence  of 
which  it  would  be  drawn  nearer  the  sun,  and  perform  its  revolu- 


272  ASTEONOMY. 

tion  in  less  time.  It  appears  to  follow  from  this  hypothesis  that, 
after  the  lapse  of  many  ages,  not  only  this  comet,  but  other  comets, 
and  the  planets  also,  must  be  precipitated  upon  the  sun. 

512.  Objections  to  Encke's  hypothesis. — It  has  been  objected  to 
Encke's  hypothesis  that  no  indication  of  a  resisting  medium  has 
been  detected  in  the  motion  of  other  comets.     Encke's  answer  to 
this  objection  is  that,  m  order  to  decide  whether  a  comet  is  affect- 
ed by  a  resisting  medium,  we  require  observations  at  three  suc- 
cessive returns  of  the  comet  to  the  sun.     Now  there  are  only 
three  comets  for  which  we  have  such  observations,  viz.,  Halley's, 
Encke's,  and  Faye's.     We  do  not  know  whether  Halley's  comet 
experiences  resistance  or  not,  because  the  disturbing  influence  of 
the  planets  through  an  entire  revolution  has  never  yet  been  com- 
puted with  sufficient  precision.    Faye's  comet,  will  be  described  in 
Arts.  515, 516. 

Biela  s  Comet. 

513.  In  1826,  Captain  Biela,  an  Austrian  officer,  discovered  a 
comet,  which  was  afterward  observed  by  other  astronomers.    The 
path  which  it  pursued  was  found  to  be  similar  to  that  of  comets 
vrhich  had  appeared  in  1772  and  1805 ;  and  Biela  concluded  that 
this  body  revolved  in  an  elliptic  orbit,  with  a  period  of  about  63- 
years. 

This  comet  has  since  been  observed  at  three  returns  in  con- 
formity with  prediction,  viz.,  in  1832,  1846,  and  1852.  It  was 
not  seen  in  1866,  although  its  computed  position  was  favorable 
for  observation,  and  there  is  reason  to  conclude  that  this  comet 
is  permanently  lost  to  our  view. 

At  perihelion  the  distance  of  this  comet  from  the  sun  is  a  little 
less  than  that  of  the  earth,  while  at  aphelion  its  distance  some- 
what exceeds  that  of  Jupiter. 

The  orbit  of  this  comet  approaches  the  earth's  orbit  within  a 
distance  less  than  the  sum  of  the  semi-diameters  of  the  earth  and 
comet  The  earth  passes  this  point  of  its  orbit  on  the  30th  of 
November.  If  Biela's  comet  should  ever  arrive  at  the  same  point 
on  the  30th  of  November,  the  earth  must  penetrate  a  portion  of 
the  comet.  In  1832,  the  comet  passed  this  point  on  the  29th  of 
October — a  circumstance  which  created  no  little  alarm. 


COMETS.  273- 

514.  Division  of  Bields  comet  into  two  comets. — In  1846,  this 
comet  presented  the  singular  phenomenon  of  a  double  comet,  OF 
two  distinct  comets  moving  through  space  side  by  side.     See 
'Plate  V.,  Fig.  3.    At  first,  one  portion  was  extremely  faint  as  com- 
'pared  with  the  other,  but  the  fainter  gradually  increased,  and  by 
the  middle  of  February  they  were  nearly  equal  in  brightness ; 
after  which  the  variable  comet  began  to  diminish,  and  in  about 
a  month  disappeared,  while  the  other  continued  visible  several 
weeks  longer  as  a  single  comet.     The  orbits  of  these  two  bodies 
were  found  to  be  ellipses  entirely  independent  of  each  other ;  and 
during  their  entire  visibility  in  1846,  their  distance  apart  was 
about  200,000  miles. 

Biela's  comet  reappeared  in  August,  1852,  and  continued  visi- 
ble about  four  weeks.  The  changes  of  relative  brilliancy  of  the 
two  comets  were  similar  to  those  observed  in  1846.  At  first  one 
body  was  fainter  than  the  other;  subsequently  the  fainter  became 
the  brightest ;  and,  a  few  days  later,  it  again  became  the  fainter 
of  the  two.  The  distance  of  the  two  bodies  from  each  other  in 
1852  was  about  1,500,000  miles.  It  has  been  found  by  compu- 
tation that  near  the  close  of  December,  1845,  Biela's  comet  passed 
extremely  near  and  probably  through  the  stream  of  November 
meteors,  Art.  529.  It  has  been  conjectured  that  this  collision 
may  have  produced  the  separation  of  this  comet  into  two  parts; 
and  that  by  subsequent  encounters  in  1859  and  1866  it  may  have 
been  farther  subdivided  and  dissipated,  so  as  to  have  become  en- 
tirely invisible  to  us. 

Fayt?s  Comet. 

515.  In  1843,  M.  Faye,  of  the  Paris  Observatory,  discovered  .1 
comet,  and  determined  its  orbit  to  be  an  ellipse,  with  a  period  of 
only  7£  years.    Le  Verrier  computed  its  orbit  with  great  care,  and 
predicted  its  succeeding  return  to  perihelion  for  April  3, 1851. 
The  comet  was  first  seen  November  28, 1850,  very  nearly  in  the 
place  assigned  it  by  Le  Verrier ;  and  it  reached  its  perihelion 
within  about  a  day  of  the  time  predicted. 

The  distance  of  Faye's  comet  from  the  sun  at  perihelion  is  161 
millions  of  miles,  and  at  aphelion  565  millions.  This  comet  is 
remarkable  as  having  an  orbit  more  closely  resembling  in  form 
the  orbits  of  the  planets  than  any  other  cometary  orbit  known,  its 
eccentricity  being  only  0.55. 

S 


274  ASTRONOMY. 

516.  Does  Faye's  comet  afford  evidence  of  a  resisting  medium? — 
The  observations  of  Faye's  comet  at  its  first  appearance  embraced 
a  period  of  nearly  six  months,  and  enabled  astronomers  to  com- 
pute the  orbit  with  uncommon  precision.     At  its  second  appear- 
ance, the  observations  embraced  a  period  of  more  than  three 
months,  and  at  its  third  appearance,  in  1858,  they  embraced  a  pe- 
riod of  more  than  one  month.     All  these  observations  may  be 
very  accurately  represented  by  an  elliptic  orbit,  without  supposing 
that  the  comet  has  experienced  any  resistance  from  an  assumed 
ether.     The  comet  made  its  fourth  appearance  in  1865,  and  its 
observed  positions  agreed  almost  exactly  with  those  which  had 
been  predicted,  showing  that  this  body  does  not  encounter  any 
appreciable  resistance. 

Encke's  comet  is  therefore  the  only  body  at  present  known 
which  requires  us  to  admit  the  existence  of  a  resisting  medium ; 
and  according  to  Professor  Encke,  this  resistance  is  not  apprecia- 
ble beyond  the  orbit  of  Venus,  and  the  density  of  the  medium  is 
assumed  to  vary  inversely  as  the  square  of  the  distance  from  the 
sun.  A  resisting  medium  must  produce  an  effect  upon  the  mo- 
tion of  a  comet,  quite  different  from  that  which  some  persons 
would  anticipate.  Such  a  medium  would  diminish  the  comet's 
tangential  velocity;  that  is,  it  would  diminish  its  centrifugal  force; 
in  consequence  of  which,  the  comet  must  be  drawn  nearer  to  the 
sun,  so  that  it  would  describe  a  smaller  orbit.  But,  according  to 
Art.  245,  when  the  orbit  diminishes,  the  absolute  velocity  increases. 
Hence  we  conclude,  that  the  absolute  velocity  of  a  planetary  or 
cometary  body  is  increased  by  encountering  a  resisting  medium. 

Brorseri's  Comet. 

517.  In  1846,  Mr.  Brorsen,  of  Denmark,  discovered  a  telescopic 
cornet,  which  has  been  found  to  revolve  around  the  sun  in  about 
5£  years.     The  date  of  its  next  arrival  at  perihelion  was  fixed  for 
September,  1851.    Its  position  at  that  time  was  very  unfavorable 
for  observations,  and  the  comet  was  not  found.    It  was,  however, 
seen  at  its  subsequent  return  to  perihelion  in  1857.     It  was  dis« 
covered  at  Berlin  March  18th,  and  passed  its  perihelion  March 
29th,  1857. 

The  distance  of  this  comet  from  the  sun  at  perihelion  is  62 
millions  of  miles,  being  less  than  the  distance  of  Venus;  and  at 
aphelion  538  millions,  which  is  somewhat  greater  than  the  dis- 


COMETS.  275 

tance  of  Jupiter.  Its  periodic  time  is  2031  days.  The  orbit  of 
this  comet,  when  projected  on  the  ecliptic,  is  included  wholly 
•within  that  of  Biela. 

This  comet  was  not  seen  in  1862  on  account  of  its  unfavorable 
position,  but  upon  its  subsequent  return  to  perihelion  in  1868  it 
was  observed  within  one  degree  of  the  place  previously  computed. 

D 'Arrest's  Comet. 

518.  In  1851,  Dr.  D' Arrest,  of  Leipsic,  discovered  a  faint  tel- 
escopic comet,  whose  orbit  was  computed  to  be  an  ellipse,  having 
a  period  of  6.4  years.    It  was  accordingly  predicted  that  it  would 
return  again  to  the  sun  about  the  last  of  November,  1857.     On 
account  of  its  great  southern  declination,  this  comet  was  not  visi- 
ble in  the  northern  hemisphere,  but  it  was  discovered  at  the  Cape 
of  Good  Hope  in  December,  1857,  and  followed  until  the  middle 
of  January.     It  passed  the  perihelion  November  28th,  and  pur- 
sued almost  exactly  the  path  predicted  for  it  in  1851.     Its  dis- 
tance from  the  sun  at  perihelion  is  111  millions  of  miles,  and  at 
aphelion  546  millions. 

WinnecJce's  Comet  and  Tuttk's  Comet. 

519.  In  1819,  M.  Pons,  at  Marseilles,  discovered  a  comet,  which 
he  continued  to  observe  for  38  days.     Its  orbit  was  computed  by 
Encke  to  be  an  ellipse,  with  a  period  of  5.6  years.     This  comet 
was  not  seen  again  until  1858,  when  it  was  rediscovered  by  Dr. 
Winnecke,  at  Bonn,  having  made  seven  revolutions  since  its  ap- 
parition in  1819,  making  the  time  of  one  revolution  5.54  years. 
Its  distance  from  the  sun  at  perihelion  is  73  millions  of  miles,  and 
at  aphelion  526  millions.     Owing  to  its  unfavorable  position,  this 
comet  was  not  seen  in  1863,  but  it  was  seen  again  in  the  summer 
of  1869,  pursuing  very  nearly  the  path  predicted. 

In  1858  a  small  comet  was  discovered  at  Cambridge,  Mass.,  by 
Mr.Tuttle,  and  was  observed  for  eleven  weeks.  Its  orbit  was 
elliptical,  and  its  elements  were  almost  identical  with  those  of  a 
comet  which  had  been  observed  in  1790,  and  it  was  predicted  that 
it  would  return  again  to  perihelion  Nov.  30th,  1871.  In  1871  the 
comet  was  found  very  near  the  place  computed  for  it,  and  thus 
its  periodical  character  was  established.  Its  perihelion  distance 
is  a  little  greater  than  that  of  the  earth ;  its  aphelion  distance 
greater  than  that  of  Saturn ;  and  its  period  13.64  years. 


276  ASTRONOMY. 

The  Comet  of  17 '44. 

520.  The  comet  of  1744  was  the  most  splendid  comet  of  the 
18th  century.     Its  distance  from  the  sun  at  perihelion  was  only 
about  one  fifth  that  of  the  earth,  or  a  little  more  than  one  half  the 
mean  distance  of  Mercury.     Three  weeks  before  the  perihelion 
passage,  its  light  was  equal  to  that  of  Jupiter  at  his  greatest  bril- 
liancy, and  a  fortnight  before  perihelion  its  light  was  little  infe- 
rior to  Yenus.     On  the  day  of  perihelion  passage  the  head  was 
seen  with  a  telescope  at  noonday,  and  many  persons  followed  it 
with  the  naked  eye  some  time  after  the  sun  had  risen. 

The  tail  of  this  comet  attained  a  length  of  19  millions  of  miles. 
A  fortnight  before  perihelion  the  tail  appeared  divided  into  two 
branches,  one  7°  and  the  other  24°  long.  On  the  day  before  peri- 
helion the  tail  exhibited  remarkable  curvature,  being  nearly  in 
the  form  of  a  semi -parabola.  Then  followed  a  week  of  cloudy 
weather,  during  which  the  comet  could  not  be  observed ;  but  six 
days  after  perihelion,  about  two  hours  before  sunrise,  when  the 
head  of  the  comet  was  far  below  the  horizon,  the  extremity  of  the 
tail  rose  above  the  horizon,  and  appeared  spread  out  like  a  fan,  as 
shown  in  Plate  V.,  Fig.  5.  This  portion  presented  the  appearance 
of  six  tails,  extending  from  30°  to  44°  from  the  head  of  the  comet 

The  Comet  of  1770. 

521.  The  comet  of  1770  is  remarkable  for  its  near  approach  to 
the  earth  and  Jupiter,  and  the  consequent  changes  in  the  form  of 
its  orbit.    This  comet  was  found  to  describe  an  elliptic  orbit,  with 
a  periodic  time  of  about  5£  years.     By  tracing  back  the  comet's 
path,  it  was  found  that  early  in  1767  it  was  very  near  to  Jupiter, 
the  distance  between  the  two  bodies  being  at  one  time  only  -y$th 
of  the  comet's  distance  from  the  sun,  in  which  position  the  influ- 
ence of  the  planet  must  have  been  three  times  greater  than  that  of 

thesun.     For,byArt.256,G:<7::^:|::^:l::l:3.     The 

motion  of  the  comet  at  this  part  of  the  orbit  being  nearly  in  the 
same  direction  as  that  of  Jupiter,  it  was  subjected  for  several 
months  to  a  powerful  disturbance  from  that  planet ;  and  the  small 
ellipse  in  which  the  comet  was  seen  to  move  in  1770  was  the  re- 
sult of  Jupiter's  attraction.  Previous  to  that  time  it  had  been 
moving  in  an  orbit  requiring  48  years  for  a  revolution,  and  its 


COMETS. 


277 


perihelion  distance  was  about  300  millions  of  miles,  at  which  dis- 
tance it  could  never  be  seen  from  the  earth. 

This  comet  has  not  been  seen  since  1770.  Its  observation  on 
its  first  return  in  1776  was  rendered  impossible  by  its  great  dis> 
tance  from  the  earth,  and  before  another  revolution  could  be  ac- 
complished it  again  passed  very  near  to  Jupiter.  In  August, 
1779,  the  distance  of  the  comet  from  Jupiter  was  only  T|T  of  its 
distance  from  the  sun,  in  which  position  the  action  of  the  planet 

must  have  exceeded  that  of  the  sun  230  times.     For  G  :  g  :  : 


:!::!:  230.  In  consequence  of  this  attraction,  the  orbit  was  so 
changed  that  the  time  of  revolution  became  16  years,  and  its  peri- 
helion distance  again  became  about  300  millions  of  miles,  at  which 
distance  the  comet  can  not  be  seen  from  the  earth.  Thus  this 
comet  has  been  entirely  invisible  from  the  earth  both  before  and 
since  the  year  1770.  The  annexed  diagram  shows  the  form  of 
the  orbit  of  this  comet  in  1770,  and  its  relation  to  the  orbits  of  the 
earth  and  Jupiter. 

Fig.  121. 


522.  Mass  of  this  comet— In  July,  1770,  this  comet  made  a  nearer 
upproach  to  the  earth  than  any  other  comet  on  record,  its  distance 
at  one  time  being  only  1,400,000  miles.  In  this  position,  the  nebu- 
losity surrounding  the  nucleus  subtended  an  angle  of  2°  23',  or 
nearly  five  times  that  of  the  moon.  Laplace  has  computed  that 
if  the  mass  of  this  comet  had  been  equal  to  that  of  the  earth,  it 
would  have  changed  the  earth's  orbit  to  such  an  extent  as  to  have 


278  ASTRONOMY. 

increased  the  length  of  the  year  by  2h.  48m.  But  it  is  proved 
from  astronomical  observations  that  the  length  of  the  year  has 
not  been  increased  by  a  quantity  so  large  as  two  seconds,  from 
which  it  is  inferred  that  the  mass  of  the  comet  can  not  have  been 
so  great  as  -njVo^  °f  the  mass  of  the  earth. 

The  mass  of  the  comet  must  indeed  have  been  smaller  than 
this,  for,  although  the  comet  approached  Jupiter  within  a  distance 
less  than  that  of  his  fourth  satellite,  the  motions  of  the  satellites 
suffered  no  perceptible  derangement 

The  Great  Comet  0/1843. 

523.  One  of  the  most  brilliant  comets  of  the  present  century 
was  the  great  comet  of  1843.     It  was  seen  in  many  parts  of  the 
world  on  the  28th  of  February,  at  midday,  close  to  the  sun ;  and 
soon  after  this,  it  became  visible  as  a  very  conspicuous  object  in 
the  evening  twilight.    The  apparent  length  of  its  tail  varied  from 
50  to  70  degrees,  and  its  greatest  real  length  was  about  120  mil- 
lions of  miles.     At  perihelion  this  comet  came  almost  in  contact 
with  the  sun's  disc,  and  it  has  been  computed  that  it  must  have 
become  2000  times  hotter  than  red-hot  iron.     For  several  days 
after  perihelion  the  tail  exhibited  a  decided  fiery  appearance. 
The  heat  to  which  it  had  been  subjected  was  doubtless  the  cause 
of  its  extraordinary  tail,  which  not  merely  attained  an  enormous 
length,  but  was  formed  with  astonishing  rapidity. 

This  comet  moved  in  a  very  elongated  ellipse.  Attempts  have 
been  made  to  identify  it  with  comets  which  appeared  in  1668  and 
1689 ;  but  the  most  careful  computations  indicate  that  its  period 
amounts  to  about  170  years. 

Donates  Comet  of  1858. 

524.  This  comet  was  discovered  at  Florence  by  Donati  in  June, 
1858,  and  for  two  months  remained  a  faint  object,  not  discernible 
by  the  unaided  eye.     During  the  latter  part  of  August,  traces  of 
a  tail  were  noticed.    The  comet  passed  through  perihelion  on  Sep- 
tember 29th,  and  was  at  its  least  distance  from  the  earth  on  Oc- 
tober 10th.     The  tail  continually  increased  until  October  10th, 
when  it  had  attained  a  length  of  50  millions  of  miles,  and  sub- 
tended an  angle  of  60°.     The  nucleus  of  the  comet  was  uncom- 
monly large,  and  was  intensely  brilliant.     It  was  not  seen  in  Eu- 
rope after  the  end  of  October,  but  in  the  southern  hemisphere  it 


SHOOTIXG  STARS.  279 

was  followed  till  March,  1859.  This  comet  undoubtedly  revolves 
in  an  elliptic  orbit,  but  the  period  can  not  be  less  than  1600  years, 
and  is  probably  about  2100  years.  This  comet  is  remarkable  for 
the  changes  which  were  noticed  in  the  number  and  dimensions 
of  its  nebulous  envelopes,  which  were  similar  to  those  described 
in  Art.  491. 

525.  Is  it  possible  for  a  comet  to  strike  the  earth? — Since  comets 
move  through  the  planetary  spaces  in  every  direction,  it  is  quite 
possible  that  in  the  lapse  of  time  the  earth  may  come  in  collision 
with  one  of  them.    The  comet  of  1770  approached  within  1,400,000 
miles  of  the  earth.    In  1832,  Biela's  comet  approached  the  earth's 
orbit  so  near  that  a  portion  of  the  orbit  must  have  been  included 
within  the  nebulosity  of  the  comet ;  the  earth  was,  however,  at 
that  time  distant  many  millions  of  miles  from  the  comet.     The 
first  comet  of  1864  also  approached  within  60(^000  miles  of  the 
earth's  orbit.     The  consequences  which  would  result  from  a  col- 
lision between  the  earth  and  a  comet  would  depend  mainly  upon 
the  mass  of  the  comet.     If  the  comet  had  no  solid  nucleus,  it  is 
probable  that  it  would  be  entirely  arrested  by  the  earth's  atmos- 
phere, and  no  portion  of  it  might  reach  the  earth's  surface. 

That  the  earth  may  some  time  pass  through  the  tail  of  a  comet 
is  highly  probable;  and,  indeed,  we  know  of  several  cases  in 
which  the  earth  has  passed  ver}r  near  to  the  tail  of  a  comet,  if  it 
has  not  been  actually  enveloped  in  the  nebulosity. 

Shooting  Stars. 

526.  Shooting  stars  aro  those  small  luminous  bodies  which  at 
night  are  frequently  seen  to  shoot  rapidly  across  the  heavens,  and 
suddenly  disappear.     They  may  be  seen  on  every  clear  night,  and 
at  times  follow  each  other  so  rapidly  that  it  is  quite  impossible 
to  count  them.     They  generally  increase  in  frequency  from  the 
evening  twilight  throughout  the  night  until  the  morning  twilight ; 
and,  when  the  light  of  day  does  not  interfere,  they  are  most 
numerous  about  6  A.M. 

527.  Height,  velocity,  etc.— By  means  of  simultaneous  observa- 
tions made  at  two  or  more  stations  at  suitable  distances  from  each 
other,  we  may  determine  their  height  above  the  earth's  surface, 
the  length  of  their  paths,  and  the  velocity  of  their  motion.     It  is 


280  ASTRONOMY. 

found  that  they  begin  to  be  visible  at  an  average  height  of  74 
miles,  and  they  disappear  at  an  average  elevation  of  50  miles. 
The  average  length  of  their  visible  paths  is  42  miles.  The  aver- 
age velocity  relative  to  the  earth's  surface  for  the  brighter  class 
of  shooting  stars  amounts  to  29  miles  per  second ;  and  they  come 
in  the  greatest  numbers  from  that  point  of  space  toward  which  the 
earth  is  moving  in  its  annual  course  around  the  sun. 

528.  The  meteors  of  August  and  November. — Shooting  stars  are 
most  numerous  in  the  month  of  August;  and  about  the  10th  of 
August  the  number  is  five  times  as  great  as  the  average  for  the 
entire  year.     The  paths  of  most  of  them  then  diverge  from  the 
constellation  Perseus,  a  region  about  40°  north  of  that  point  to- 
ward which  the  earth  at  that  time  is  moving. 

In  the  year  1833,  shooting  stars  appeared  in  extraordinary  num- 
bers on  the  morfting  of  November  13th.  It  was  estimated  that 
the  number  visible  at  a  single  station  could  not  have  been  less 
than  200,000.  They  seemed  to  emanate  chiefly  from  a  point  in 
the  constellation  Leo,  which  is  about  10°  north  of  that  point  in 
the  heavens  toward  which  the  earth  at  that  time  was  moving.  A 
similar  exhibition  took  place  on  the  12th  of  November,  1799,  ns 
also  on  several  other  years  about  the  same  day  of  November. 

A  brilliant  display  of  meteors  was  observed  in  Europe  on  the 
morning  of  November  14th,  1866,  and  again  in  the  United  States 
on  the  morning  of  November  14th,  1867,  and  also  in  1868. 

529.  Meteoric  orbits,  etc. — Ilaving  determined  the  velocity  and 
direction  of  a  meteor's  path  with  reference  to  the  earth,  we  can 
compute  the  direction  and  velocity  of  the  motion  with  reference 
to  the  sun.     In  this  manner  it  has  been  shown  that  these  bodies, 
before  they  approached  the  earth,  were  revolving  about  the  sun 
in  ellipses  of  considerable  eccentricity.     In  some  instances  the 
velocity  has  been  so  great  as  to  indicate  that  the  path  differed  lit- 
tle from  a  parabola. 

Thus  we  see  that  ordinary  shooting  stars  are  bodies  moving 
through  space  in  paths  similar  to  the  comets;  and  it  is  probable 
that  they  do  not  differ  materially  from  comets  except  in  their  di- 
mensions, and  perhaps  also  in  their  density. 

Their  light  probably  results  from  the  heat  generated  by  the 
compression  of  the  air  before  them.  It  has  been  objected  that  at 


DETONATING   METEORS.  281 

the  height  of  50  miles  the  atmosphere  is  too  rare  to  develop  so 
much  heat.  But  we  know  that  the  motion  of  a  large  body  mov- 
ing about  30  miles  per  second  is  entirely  lost  in  a  second  or  two, 
and  this  motion,  communicated  to  the  particles  of  the  surround- 
ing air,  must  be  sufficient  to  develop  an  enormous  amount  of 
heat  and  light. 

We  also  conclude  that  shooting  stars  are  not  distributed  uni- 
formly through  space,  but  many  of  them  are  grouped  together, 
forming  complete  or  incomplete  rings  of  minute  bodies  revolving 
together  around  the  sun.  These  rings  are  so  situated  that  the 
earth  encounters  one  of  them  annually  on  the  10th  of  August, 
and  another  occasionally  on  the  12th  of  November,  furnishing 
meteoric  displays  of  unusual  splendor.  The  plane  of  the  August 
zone  appears  to  be  nearly  perpendicular  to  the  plane  of  the  earth's 
orbit ;  and  this  is  the  reason  why  at  that  time  the  radiant  point 
is  found  so  far  distant  from  the  ecliptic. 

Detonating  Meteors. 

530.  O i-d i nary  shooting  stars  are  not  accompanied  by  any  audi- 
ble sound,  though  sometimes  seen  to  break  into  pieces.  Occa- 
sionally meteors  of  extraordinary  brilliancy  are  succeeded  by  an 
explosive  noise.  These  have  been  called  detonating  meteors.  On 
the  morning  of  November  loth,  1859,  a  meteor  passed  over  the 
southern  part  of  New  Jersey,  and  was  so  brilliant  that  its  flash 
attracted  attention  in  the  presence  of  an  unclouded  sun.  Soon 
after  the  flash,  there  was  heard  a  series  of  terrific  explosions, 
which  were  compared  to  the  discharge  of  a  thousand  cannon. 
From  a  comparison  of  numerous  observations,  it  was  computed 
that  the  height  of  this  meteor  when  first  seen  was  over  60  miles  ; 
and  when  it  exploded  its  height  was  20  miles.  The  length  of  its 
visible  path  was  more  than  40  miles.  Its  velocity  relative  to  the 
earth  was  at  least  20  miles  per  second ;  but  its  velocity  relative 
to  the  sun  was  about  28  miles  per  second,  indicating  that  it  was 
moving  about  the  sun  in  a  very  eccentric  ellipse,  or  perhaps  a 
parabola. 

On  the  2d  of  August,  1860,  in  the  evening,  a  magnificent  fire- 
ball was  seen  throughout  the  whole  region  from  Pittsburg  to  New 
Orleans,  and  from  Charleston  to  St.  Louis.  A  few  minutes  after 
its  disappearance  there  was  heard  a  tremendous  explosion  like 
the  sound  of  distant  cannon.  The  length  of  its  visible  path  was 


282  ASTRONOMY. 

about  240  miles,  and  its  time  of  flight  was  8  seconds,  showing  a 
velocity  of  30  miles  per  second.  Its  velocity  relative  to  the  sun 
was  24  miles  per  second. 

531.  Number,  velocity,  etc. — The  number  of  detonating  meteors 
recorded  m  scientific  journals  is  over  800.     Their  average  height 
at  the  first  instant  of  apparition  is  92  miles,  and  at  the  instant  of 
vanishing  is  32  miles.     Their  average  velocity  is  19  miles  per 
second. 

Comparing  these  results  with  those  derived  from  the  ordinary 
shooting  stars,  we  conclude  that  the  two  classes  of  bodies  do  not 
probably  differ  much  from  each  other  except  in  size  and  density. 
The  noise  which  succeeds  the  appearance  of  a  detonating  meteor 
is  perhaps  due  to  the  collapse  of  the  air  rushing  into  the  vacuum 
which  is  left  behind  the  advancing  meteor.  No  audible  sound 
proceeds  from  ordinary  shooting  stars,  because  they  are  small 
bodies,  of  feeble  density,  and  are  generally  consumed  while  yet 
at  an  elevation  of  50  miles  above  the  earth's  surface. 

Aerolites. 

532.  There  is  no  evidence  that  any  thing  coming  from  ordinary 
shooting  stars  ever  reaches  the  earth's  surface ;  but  occasionally 
solid  bodies  descend  to  the  earth's  surface  from  beyond  the  earth's 
atmosphere.     These  are  called  aerolites.     In  December,  1807,  a 
meteor  of  great  brilliancy  passed  over  the  southern  part  of  Con- 
necticut, and  soon  after  its  disappearance  there  were  heard  three 
loud  explosions  like  those  of  a  cannon,  and  there  fell  a  shower  of 
meteoric  stones.     The  entire  weight  of  all  the  fragments  discov- 
ered was  at  least  300  pounds.    The  specific  gravity  of  these  stones 
was  3.6;  their  composition  was  one  half  silex,  one  third  oxyd  of 
iron,  and  the  remainder  chiefly  magnesia.    The  length  of  the  vis- 
ible path  of  this  meteor  was  at  least  100  miles,  and  its  velocity 
several  miles  per  second. 

In  May,  1860,  an  aerolite  exploded  over  Eastern  Ohio,  and  from 
it  descended  a  shower  of  stones  whose  entire  weight  was  esti- 
mated at  700  pounds.  Their  specific  gravity  was  3.54,  and  their 
composition  very  similar  to  that  of  the  meteor  of  1807.  There 
are  20  well-authenticated  cases  in  which  aerolites  have  fallen  in 
the  United  States  since  1807.  The  specific  gravity  of  16  of  these 
meteors  ranged  from  3  to  3.66. 


AEROLITES.  283 

In  July,  1847,  an  aerolite  exploded  over  Bohemia,  and  from  it 
there  were  seen  to  descend  two  masses  of  iron,  which  together 
weighed  72  pounds.  Its  specific  gravity  was  7.71.  Its  compo- 
sition was  92  per  cent,  of  iron,  5  per  cent,  nickel,  with  a  small 
quantity  of  cobalt,  etc.  There  are  one  or  two  other  cases  in  which 
iron  meteors  have  been  known  to  fall  to  the  earth ;  and  there  have 
been  found  over  100  other  similar  masses  believed  to  be  aerolites, 
although  the  date  of  their  fall  is  unknown. 

The  elements  of  which  aerolites  consist  are  the  same  as  those 
found  in  the  crust  of  the  earth ;  yet  the  manner  in  which  these 
elements  are  combined  is  peculiar,  so  that  the  general  aspect  of 
aerolites  is  sufficient  to  distinguish  them  from  all  terrestrial  min- 
erals. 

533.  Origin  of  aerolites. — Various  hypotheses  have  been  pro- 
posed to  account  for  the  origin  of  aerolites. 

1st.  It  has  been  conjectured  that  they  are  formed  in  the  atmos- 
phere like  rain  or  hail.  This  supposition  is  inadmissible,  because, 
allowing  the  aerolite  to  be  once  formed,  there  is  no  known  cause 
which  could  impel  it  in  a  direction  nearly  horizontal  with  a  ve- 
locity of  several  miles  per  second. 

2d.  It  has  been  conjectured  that  aerolites  are  masses  ejected 
from  terrestrial  volcanoes.  This  supposition  is  inadmissible,  be- 
cause the  greatest  velocity  with  which  stones  have  ever  been 
ejected  from  volcanoes  is  less  than  two  miles  per  second,  and  the 
direction  of  this  motion  must  be  nearly  vertical ;  while  aerolites 
frequently  move  in  a  direction  nearly  horizontal,  and  with  a  ve- 
locity of  several  miles  per  second. 

3d.  It  has  been  conjectured  that  aerolites  have  been  ejected  from 
volcanoes  in  the  moon  with  a  velocity  sufficient  to  carry  them  out 
of  the  sphere  of  the  moon's  attraction  into  that  of  the  earth's  at- 
traction. This  supposition  is  unsatisfactory,  because  the  lunar 
volcanoes  are  at  present  entirely  extinct.  If,  then,  aerolites  once 
belonged  to  the  moon,  they  must  have  been  projected  from  its 
surface  many  years  ago.  Since  that  time  they  must  have  been 
moving  in  orbits  around  some  larger  body,  such  as  the  earth  or 
the  sun  ;  that  is,  whatever  may  have  been  the  first  source  of  aero- 
lites, they  must  now  be  regarded  as  satellites  of  the  earth  or  the 
sun. 


284  ASTRONOMY. 

534.  Orbits  of  aerolites. — The  facts  which  have  been  established 
respecting  shooting  stars  and  detonating  meteors  can  leave  but 
little  doubt  that  aerolites  are  bodies  revolving  about  the  sun  like 
the  planets  and  comets,  and  are  encountered  by  the  earth  in  its 
annual  motion  around  the  sun  ;  and  it  is  probable  that  the  chief 
difference  between  these  three  classes  of  bodies  depends  upon  their 
size  and  density. 

We  hence  conclude  that  the  interplanetary  spaces,  instead  of 
being  absolutely  void,  are  filled  with  a  countless  number  of  mi- 
nute bodies,  whose  aggregate  mass  must  be  very  great  The  com- 
ets, like  the  earth,  must  encounter  an  immense  number  of  these 
bodies,  and  a  part  of  their  motion  must  be  thereby  destroyed. 
This  effect  may  be  appreciable  in  the  case  of  the  periodic  comets, 
although  it  is  thus  far  inappreciable  in  the  case  of  the  earth  and 
the  other  planets. 


CHAPTER  XIX. 

THE   FIXED  STARS — THEIR  LIGHT,  THEIR  DISTANCE,  AND 
THEIR   MOTIONS. 

535.  What  is  a  fixed  star  ? — The  fixed  stars  are  so  called  because 
from  century  to  century  they  preserve  almost  exactly  the  same 
positions  with  respect  to  each  other.  Many  of  the  stars  form 
groups  which  are  so  peculiar  that  they  are  easily  identified ;  and 
the  relative  positions  of  these  stars  are  nearly  the  same  now  as 
they  were  two  thousand  years  ago.  Accurate  observations,  how- 
ever, made  with  telescopes,  have  proved  that  many,  and  probably 
all  of  the  so-called  fixed  stars,  have  a  real  motion.  There  are, 
however,  only  about  30  stars  whose  motion  is  as  great  as  one  sec- 
ond in  a  year,  and  generally  the  motion  is  only  a  few  seconds  in 
a  century. 

530.  How  the  fixed  stars  are  classified. — The  stars  are  divided 
into  classes  according  to  their  different  degrees  of  apparent  bright- 
ness. The  most  conspicuous  are  termed  stars  of  the  first  magni- 
tude ;  those  which  are  next  in  order  of  brightness  are  called  stars 
of  the  second  magnitude,  and  so  on,  the  first  six  magnitudes  em- 
bracing all  which  can  be  distinctly  located  by  the  naked  eye 


THE   FIXED   STARS.  285 

Telescopic  stars  are  classified  in  a  similar  manner  down  to  the 
twelfth,  and  even  smaller  magnitudes. 

The  distribution  of  stars  into  magnitudes  is  arbitrary,  and  as- 
tronomers have  differed  in  the  magnitude  they  have  assigned  to 
the  same  star.  According  to  the  best  authority,  the  number  of 
stars  of  the  first  magnitude  is  20 ;  of  the  second  magnitude,  34 ; 
third,  141;  fourth,  327;  fifth,  959;  and  sixth,  4424;  making  5905 
stars  visible  to  the  naked  eye.  Of  these  only  about  one  half  can 
be  above  the  horizon  at  one  time ;  and  it  is  only  on  the  most  fa- 
vorable nights  that  stars  of  the  sixth  magnitude  can  be  clearly 
distinguished  by  the  naked  eye.  Even  then,  only  the  brighter 
stars  can  be  seen  near  the  horizon. 

The  number  of  stars  of  the  seventh  magnitude  is  estimated  at 
13,000 ;  eighth  magnitude,  40,000 ;  and  ninth  magnitude,  142,000 ; 
making  about  200,000  stars  from  the  first  to  the  ninth  magnitude. 
It  is  estimated  that  the  number  of  stars  visible  in  Herschel's  re- 
flecting telescope  of  18  inches  aperture  was  more  than  20  mil- 
lions ;  and  the  number  visible  in  more  powerful  telescopes  is  still 
greater. 

537.  Comparison  of  the  brightness  of  the  stars. — Sir  W.  Ilerschel 
estimated  that  if  an  average  star  of  the  sixth  magnitude  be  taken 
as  unity,  the  light  emitted  by  an  average  star  of  the  fifth  magni- 
tude will  be  represented  by  2 ;  one  of  the  fourth  magnitude  by 
6 ;  of  the  third  magnitude  by  12 ;  the  second  magnitude  by  25 ; 
and  the  first  magnitude  by  100.     There  is,  however,  considerable 
variety  in  the  brightness  of  stars  that  are  classed  as  of  the  same 
magnitude.    The  light  of  Sirius,  the  brightest  star  in  the  heavens, 
is  from  5  to  10  times  as  great  as  some  of  the  stars  of  the  first 
magnitude,  and  more  than  300  times  as  great  as  an  average  star 
of  the  sixth  magnitude. 

538.  Cause  of  this  diversity  of  brightness. — It  is  probable  that 
these  varieties  of  magnitude  are  chiefly  caused  by  difference  of 
distance  rather  than  by  difference  of  intrinsic  splendor  among 
the  objects  themselves.    Those  stars  which  are  placed  immediate- 
ly about  our  solar  system  appear  bright  in  consequence  of  their 
proximity,  and  are  called  stars  of  the  first  magnitude;  those  which 
lie  beyond  are  more  numerous,  and  appear  less  bright,  and  are 
called  stars  of  the  second  magnitude ;  and  thus,  as  the  distance 


286  ASTRONOMY. 

of  the  stars  increases,  their  apparent  brightness  diminishes,  until 
at  a  certain  distance  they  become  invisible  to  the  naked  eya 

Some  deviations  from  this  general  rule  are  to  be  expected.  In 
fact,  some  of  the  fainter  stars  are  among  those  which  are  nearest 
to  us. 

539.  Have  the  fixed  stars  a  sensible  disc? — When  a  telescope  is 
directed  to  a  planet,  the  planet  appears  with  a  distinct  disc,  like 
that  which  the  moon  presents  to  the  naked  eye.     But  it  is  differ- 
ent even  with  the  brightest  of  the  fixed  stars.     The  telescope,  in- 
stead of  magnifying,  actually  diminishes  them.    A  star  viewed  by 
the  naked  eye  appears  surrounded  by  a  radiation,  and  the  ap- 
pearance may  be  represented  by  a  dot  with  rays  diverging  from 
every  side  of  it.    The  telescope  cuts  off  this  radiation,  and  exhib- 
its the  star  as  a  lucid  point  of  very  small  diameter,  even  when 
the  highest  magnifying  powers  are  employed.     With  a  power  of 
6000,  the  apparent  diameter  of  the  stars  seems  less  than  with  lower 
powers. 

The  brighter  stars,  when  viewed  with  the  best  telescope,  do, 
however,  exhibit  a  small  disc ;  but  this  disc  is  spurious,  and  prob- 
ably arises  from  the  dispersion  of  light  in  passing  through  the 
earth's  atmosphere.  That  these  discs  are  not  real  is  proved  by 
the  fact  that  they  are  not  magnified  by  an  increase  of  telescopic 
power,  and  also  by  the  fact  that,  in  the  occultation  of  a  bright  star 
by  the  moon,  its  extinction  is  absolutely  instantaneous,  not  the 
smallest  trace  of  gradual  diminution  of  light  being  perceptible. 

540.  How  do  the  stars  become  visible  to  us  ? — The  term  magnitude 
applied  to  the  stars  is  therefore  used  to  designate  simply  their 
relative  brightness.     None  of  the  stars  have  any  measurable  mag- 
nitude at  all.     There  is,  however,  reason  to  believe  that  the  ab- 
solute diameters  of  the  stars  are  very  great;  hence  we  are  com- 
pelled to  conclude  that  the  distance  of  these  bodies  is  so  enormous 
that  their  apparent  diameter  seen  from  the  earth  is  6000  times 
less  than  any  angle  which  the  naked  eye  is  capable  of  appre- 
ciating. 

Stars,  then,  become  sensible  to  the  eye,  not  by  subtending  an 
appreciable  angle,  but  from  the  intensity  of  the  light  which  they 
emit.  The  quantity  of  light  which  the  eye  receives  from  a  star 
varies  inversely  as  the  square  of  its  distance.  At  a  certain  dis- 


THE   FIXED  STABS.  287 

tance  this  light  is  insufficient  to  produce  sensation,  and  the  star 
becomes  invisible. 

When  a  star  becomes  invisible  to  the  naked  eye,  the  telescope 
may  render  it  visible  by  uniting  in  the  image  as  many  rays  as  can 
enter  the  aperture  of  the  object-glass.  The  increase  of  illumina- 
tion from  the  use  of  a  telescope  will  depend  upon  the  ratio  of  the 
area  of  the  aperture  of  the  object-glass  to  that  of  the  pupil  of  the 
eye.  By  augmenting  the  aperture  of  the  telescope  we  may  there- 
fore increase  the  apparent  brightness  of  an  object,  so  that  a  star 
of  the  sixth  magnitude  may  appear  as  bright  as  a  star  of  the  first 
magnitude  does  to  the  naked  eye. 

541.  Twinkling  of  the  stars.— The  scintillation,  or  twinkling  of 
the  stars,  which  contrasts  so  strongly  with  the  steady  light  of  the 
principal  planets,  is  an  optical  phenomenon,  supposed  to  be  due 
to  what  is  termed  the  interference  of  light.     Humboldt,  the  cele- 
brated traveler,  states  that  under  the  serene  sky  of  Cumana,  in 
South  America,  the  stars  do  not  twinkle  after  they  have  attained 
an  elevation,  on  the  average,  of  15°  above  the  horizon. 

542.  Division  into  constellations. — For  the  sake  of  more  readily 
distinguishing  the  stars,  they  have  been   divided  into  groups 
called  constellations.     These  constellations  are  represented  under 
the  forms  of  various  animals,  such  as  bears,  lions,  goats,  serpents, 
and  so  on.    In  some  instances  we  may  easily  imagine  that  the  ar- 
rangement of  the  stars  bears  some  resemblance  to  the  object  from 
which  the  constellation  is  named,  as,  for  example,  the  Swan  and 
the  Scorpion;  in  other  instances  no  such  resemblance  can  be 
traced.     This  fanciful  mode  of  grouping  the  stars  is  of  very  an- 
cient date,  and  is  continued  by  modern  astronomers  chiefly  for 
the  sake  of  avoiding  the  confusion  that  might  arise  from  an  al- 
teration in  the  old  system. 

543.  Names  of  the  constellations. — There  are  twelve  constellations 
lying  upon  the  zodiac,  and  hence  called  the  zodiacal  constellations, 
viz.,  Aries,  Taurus,  Gemini,  Cancer,  Leo,  Virgo,  Libra,  Scorpio, 
Sagittarius,  Capricornus,  Aquarius,  and  Pisces.     These  are  also 
the  names  of  the  twelve  divisions  of  30°  each  into  which  the 
ecliptic  has  been  divided ;  but  the  effect  of  precession,  which 
throws  back  the  place  of  the  equinox  among  the  stars  50"  a  year, 


288  ASTRONOMY. 

has  caused  a  displacement  of  the  signs  of  the  zodiac  with  respect 
to  the  corresponding  constellations.  The  sign  Taurus  at  present 
occupies  the  constellation  Aries,  the  sign  Gemini  the  constellation 
Taurus,  and  so  on,  the  signs  having  retreated  among  the  stars  30° 
since  the  present  division  of  the  zodiac  was  adopted. 

The  principal  constellations  in  the  northern  half  of  the  heavens, 
in  addition  to  such  of  the  zodiacal  ones  as  lie  north  of  the  celes- 
tial equator,  are : 

Andromeda.     Cassiopeia.  Draco.          Perseus. 

Aquila.  Cepheus.  Hercules.     Ursa  Major. 

Auriga,  Corona  Borealis.     Lyra.  Ursa  Minor. 

Bootes.  Cygnus.  Pegasus. 

The  principal  constellations  situated  on  the  south  side  of  the 
equator,  exclusive  of  the  six  southern  zodiacal  ones,  are: 
Argo  Navis.  Cetus.  Ophiuchus. 

Canis  Major.  Crux.  Orion. 

Canis  Minor.          Eridanus.  Piscis  Australis. 

Centaur  us.  Monoceros. 

Others  will  be  found  upon  celestial  globes  and  charts,  raising  the 
total  number  of  constellations  at  present  recognized  by  astrono- 
mers to  about  eighty. 

544  How  particular  stars  are  designated. — Many  of  the  brighter 
stars  had  proper  names  assigned  them  at  a  very  early  date,  as 
Sirius,  Arcturus,  Rigel,  Aldebaran,  etc.,  and  by  these  names  they 
are  still  commonly  distinguished. 

It  was  the  custom  in  former  times  to  indicate  the  locality  of  a 
star  by  its  position  in  the  constellation  to  which  it  belonged ;  but 
this  method  was  found  to  be  extremely  tedious,  besides  being  fre- 
quently liable  to  misconception.  Bayer,  a  German  astronomer, 
in  1604  published  a  series  of  maps  of  the  heavens,  in  which  the 
stars  of  each  constellation  were  distinguished  by  the  letters  of  the 
Greek  and  Eornan  alphabets,  the  brightest  being  called  a,  the  next 
/3,  and  so  on.  Thus  a  Lyras  denotes  the  brightest  star  in  the  con- 
stellation Lyra,  /3  Lyras  the  second  star,  and  so  on. 

In  consequence  either  of  a  want  of  proper  care  in  assigning  let- 
ters to  the  stars,  or  perhaps  from  a  real  change  of  brightness  of 
the  stars  since  the  time  of  Bayer,  we  sometimes  find  that  the 
brightness  of  the  stars  in  a  constellation  does  not  follow  the  order 
of  the  letters  by  which  they  are  distinguished.  Thus  a  Draconis 


THE   FIXED  STARS.  289 

is  not  so  bright  as  either  /3  or  y  of  the  same  constellation.  Flam- 
steed,  the  first  astronomer  royal  at  Greenwich,  distinguished  the 
stars  of  each  constellation  -by  the  numerals  1,  2,  3,  etc.,  and  stars 
are  often  referred  to  by  these  numbers.  In  large  catalogues  of 
stars,  the  stars  are  usually  numbered  continuously  from  begin- 
ning to  end  in  the  order  of  their  right  ascensions. 

545.  Remarkable  constellations  enumerated. — One  of  the  most  con- 
spicuous  constellations  in  the  northern  firmament  is  Ursa  Major, 
or  the  Great  Bear,  in  which  we  find  seven  stars  which  may  easily 
be  conceived  to  form  the  outline  of  a  dipper,  of  which  the  two 
brightest  are  nearly  in  a  straight  line  with  the  pole  star,  and  are 
hence  called  the  pointers.     They  are  not  far  from  the  zenith  at 
New  Haven  at  10  o'clock  in  the  evening  in  the  month  of  April. 

The  constellation  Cassiopeia  presents  six  stars  which  may  be 
conceived  to  form  the  outline  of  a  chair.  It  is  not  far  from  the 
zenith  at  10  o'clock  in  the  evening  in  the  month  of  October. 

The  constellation  Ursa  Minor  contains  seven  stars  which  may 
also  be  conceived  to  form,  the  outline  of  a  dipper,  the  pole  star 
forming  the  extremity  of  the  handle.  The  principal  stars  of 
these  three  constellations  are  represented  in  Fig.  2,  page  14. 

The  constellation  Orion  is  one  of  the  most  magnificent  in  the 
heavens,  and  with  some  imagination  may  be  conceived  to  resem- 
ble a  great  giant.  It  is  in  the  south  at  10  o'clock  in  the  evening 
in  the  month  of  January.  To  the  left  of  Orion,  and  a  little  below 
it,  is  then  seen  the  star  Sinus,  which  far  surpasses  all  others  in 
brilliancy. 

The  square  of  Pegasus  is  formed  by  four  moderately  bright 
stars,  which  appear  at  a  considerable  altitude  above  the  horizon 
in  th,e  southern  quarter  of  the  sky  about  10  in  the  evening  in  the 
middle  of  October. 

The  Pleiades  form  a  group  of  stars  in  the  constellation  Taurus. 
The  naked  eye  discovers  six  or  seven,  but  in  the  telescope  up- 
ward of  two  hundred  are  revealed.  This  group  passes  the  merid- 
ian at  10  o'clock  in  the  evening  in  the  month  of  December.  A 
little  below,  and  to  the  left  of  the  Pleiades,  is  a  wedge  of  stars 
called  the  Hyades,  of  which  Aldebaran  is  the  conspicuous  member. 

546.  Catalogues  of  stars.— Various  catalogues  of  stars  have  been 
formed,  in  which  are  indicated  their  right  ascensions  and  declina- 

T 


290  ASTRONOMY. 

tions  for  a  certain  epoch.  Hipparchus  is  believed  to  have  been 
the  first  who  undertook  such  a  compilation,  128  years  before  the 
Christian  era;  His  catalogue  included  1022  stars,  and  has  been 
preserved  to  us  in  the  Almagest  of  Claudius  Ptolemy.  Some 
modern  catalogues  contain  a  much  larger  number  of  stars.  The 
British  Association  catalogue  contains  8377  stars;  the  catalogue 
of  Lalande  contains  47,390  stars;  Cooper's  catalogue  contains 
60,066  stars  near  the  ecliptic ;  and  the  entire  number  tabulated 
at  the  present  time  amounts  to  several  hundreds  of  thousands. 

547.  Periodic  stars. — Some  stars  exhibit  periodical  changes  in 
their  brightness,  and  are  therefore  called  periodic  stars.  One  of 
the  most  remarkable  of  this  class  is  the  star  Omicron  Ceti,  often 
termed  Mira,  or  the  wonderful  star.  This  star  retains  its  greatest 
brightness  for  about  14  days,  being  then  usually  equal  to  a  star 
of  the  second  magnitude.  It  then  decreases,  and  in  about  two 
months  ceases  to  be  visible  to  the  naked  eye.  After  remaining 
thus  invisible  for  six  or  seven  months,  it  reappears,  and  increases 
gradually  for  two  months,  when  it  recovers  its  maximum  splen- 
dor. It  goes  through  all  its  changes  in  332  days,  and  in  1879  its 
maximum  brilliancy  occurred  on  the  llth  of  September.  At  the 
times  of  the  least  light,  it  becomes  reduced  to  the  tenth  or  twelfth 
magnitude. 

Another  remarkable  periodic  star  is  Algol,  in  the  constellation 
Perseus.  It  generally  appears  of  the  second  magnitude,  and  con- 
tinues thus  for  about  61  hours.  It  then  diminishes  in  brightness, 
and  in  less  than  four  hours  is  reduced  to  a  star  of  the  fourth  mag- 
nitude, and  thus  remains  about  twenty  minutes.  It  then  in- 
creases, and  in  about  four  hours  more  it  recovers  its  original  splen- 
dor. The  exact  period  in  which  all  these  variations  are  performed 
is  2d.  20h.  48m.  55s.  It  was  at  its  minimum  of  brightness  in 
1879,  November  21st,  at  8h.  6m.  in  the  evening,  New  Haven 
time,  from  which  data  the  time  of  any  other  minimum  can  be 
computed. 

There  are  more  than  100  stars  known  to  be  variable  to  a  great- 
er or  less  extent.  The  periods  of  these  changes  vary  from  a  few 
days  to  many  years.  The  star  34  Cygni  varies  from  the  third  to 
the  sixth  magnitude  in  a  period  of  about  18  years.  The  bright 
star  Capella,  in  the  constellation  Auriga,  is  believed  to  have  in- 
creased in  lustre  during  the  present  century,  while  within  tho 


THE  FIXED   STARS.  291 

same  period  one  of  the  seven  bright  stars  (8)  in  Ursa  Major  has 
probably  diminished.  Many  instances  of  a  similar  kind  might  be 
mentioned. 

548.  Cause  of  this  periodicity. — These  phenomena  have  been  ex- 
plained, 1st,  by  supposing  that  a  dark,  opaque  body  may  revolve 
about  the  variable  star,  and  at  certain  times  intercept  a  portion  of 
its  light;  or,  2d,  that  a  nebulous  body  of  great  extent  may  re- 
volve round  the  star,  and  intercept  a  portion  of  its  light  when  in- 
terposed between  us  and  the  star.     3d.  The  stars  themselves  may 
not  be  uniformly  luminous  all  over  their  surfaces,  but  occasion- 
ally, from  their  axial  rotations,  present  toward  the  earth  a  disc 
partially  covered  with  dark  spots,  thereby  shining  with  a  dimmer 
light.     4th.  Some  stars  may  have  the  form  of  thin  flat  discs,  and 
by  rotation  present  to  us  alternately  their  edge  and  their  flat  side, 
producing  corresponding  changes  of  brightness. 

We  have  seen,  Art.  174,  that  the  light  of  our  sun  exhibits  peri- 
odical changes ;  in  other  words,  our  sun  is  a  variable  star  with  a 
period  of  about  eleven  years.  It  seems  most  philosophical  to 
conclude  that  the  changes  in  the  periodic  stars  are  due  to  causes 
analogous  to  those  which  exist  in  our  own  system ;  and  it  is  not 
surprising  that  among  the  immense  multitude  of  stars  a  few 
should  be  found  in  which  the  changes  of  brightness  are  far 
greater  than  in  the  case  of  our  sun. 

549.  Temporary  stars. — Several  instances  are  recorded  of  stars 
Fu-ldenly  appearing  where  none  had  before  been  observed,  some- 
times surpassing  the  light  of  stars  of  the  first  magnitude,  re- 
maining thus  for  a  short  time,  and  then  gradually 'fading  away. 
The  first  on  record  was  observed  by  Hipparchus  134  B.C.,  the 
disappearance  of  which  is  said  to  have  led  that  astronomer  to 
compile  the  star-catalogue  bearing  his  name.     In  the  year  389 
A.D.  a  star  blazed  forth  near  a  Aquilo3,  which  shone  for  three 
weeks,  appearing  as  splendid  as  the  planet  Venus,  after  which  it 
disappeared,  and  has  never  since  been  seen.     In  the  autumn  of 
1572  a  new  star  suddenly  appeared  in  the  constellation  Cassiopeia. 
When  first  noticed  it  was  as  bright  as  Sirius,  the  brightest  star  in 
our  firmament;  and  it  finally  attained  such  splendor  that  it  was 
distinctly  visible  at  midday.     In  about  a  month  it  began  to  di- 
minish, and  in  sixteen  months  it  entirely  disappeared. 


292  ASTRONOMY. 

Another  temporary  star  became  suddenly  visible  in  Ophiuchus 
in  1604,  and  exceeded  Jupiter  in  splendor.  It  remained  visible 
till  1606,  and  then  disappeared. 

In  1848  a  star  of  tne  fourth  magnitude  was  seen  in  the  constel- 
lation Ophiuchus,  in  a  place  where  no  star  had  ever  been  observed 
before.  After  a  few  weeks  it  declined  in  brightness,  and  has  now 
faded  away  to  the  twelfth  magnitude,  so  that  it  can  not  be  seen 
without  a  superior  telescope.  It  is  possible  that  the  temporary 
stars  do  not  differ  from  the  periodic  stars  except  in  the  length  ot 
their  periods. 

550.  Distance  of  the  fixed  stars. — That  the  distance  of  the  fixed 
stars  from  the  earth  is  immense  is  proved  by  the  following  con- 
siderations.    The  earth,  in  its  annual  course  around  the  sun,  re- 
volves in  an  orbit  whose  diameter  is  190  millions  of  miles.     The 
station  from  which  we  observe  the  stars  on  the  1st  of  January  is 
distant  190  millions  of  miles  from  the  station  from  which  we  view 
them  on  the  1st  of  July ;  yet  from  these  two  remote  points  the 
stars  present  the  same  appearance,  proving  that  the  diameter  of 
the  earth's  orbit  must  be  a  mere  point  compared  with  the  distance 
of  the  nearest  stars. 

551.  Annual  parallax. — The  greatest  angle  which  the  radius  of 
the  earth's  orbit  subtends  at  a  fixed  star  is  called  its  annual  paral- 
lax.   Numerous  attempts  have  been  made  to  measure  the  amount 
of  this  parallax.     Suppose  a  star  to  be  situated  at  the  pole  of  the 
ecliptic,  and  that  it  is  near  enough  to  the  earth  to  have  a  sensible 

Fi"  122  parallax.     Then,  while  the  earth  travels  round 

the  sun,  the  star,  as  projected  on  the  distant  firma- 
ment, will  appear  to  describe  a  small  circle,  ABCD, 


/ "X 


J       5        ) c  waose  centre,  S,  is  on  the  line  joining  the  sun  and 
\  /      star ;  and  the  diameter  of  this  circle  will  dimin- 

ish as  the  distance  of  the  star  from  the  earth  in- 


B 

creases. 

If  the  star  is  situated  in  the  plane  of  the  ecliptic,  then,  for  three 
rig  123  months  of  the  year,  it  will  appear  to  move  a  little 

s  to  the  east  of  its  mean  position,  and  in  the  next 

°  three  months  it  will  return  to  its  first  position.  It 
will  then  appear  to  move  a  little  to  the  west  of  its  mean  position, 
and  afterward  return  to  its  first  position,  its  apparent  motion  be- 
ing confined  to  a  straight  line,  AC. 


THE   FIXED  STARS.  293 

If  the  star  is  situated  between  the  ecliptic  and  its  pole,  the  mo- 
tion of  the  earth  about  the  sun  will  give  to  the  star         Fitr 
an  apparent  motion  in  an  ellipse,  ABCD,  whose 
eccentricity  will  increase  as  the  star's  latitude  de- 
creases. 

If,  then,  a  fixed  star  had  any  considerable  paral- 
lax, it  would  be  easy  to  discover  it  by  measuring  accurately  its 
position  from  one  season  to  another ;  but,  among  the  many  thou- 
sand stars  which  have  been  carefully  observed  by  astronomers, 
not  one  has  been  found  which  exhibits  a  parallax  exceeding  one 
second. 

552.  Parallax  of  Alpha  Centauri. — Observations  made  upon  the 
star  Alpha  Centauri,  one  of  the  brightest  stars  of  the  southern 
hemisphere,  indicate  an  annual  parallax  of  -j—j-ths  of  a  second. 
Having  determined  the  parallax,  we  can  compute  the  distance  of 
the  star  by  the  proportion 

sin.  0".92  :  1 : :  95  millions  of  miles  :  the  distance  of  the  star, 
which  is  found  to  be  twenty  millions  of  millions  of  miles.  This 
distance  is  so  immense  that  a  ray  of  light,  moving  at  the  rate  of 
192,000  miles  per  second,  requires  3-^  years  to  travel  from  this 
star  to  the  earth.  We  do  not  see  the  star  as  it  actually  is,  but  it 
shines  with  the  light  emitted  3-j  years  ago.  Hence,  if  it  were 
obliterated  from  the  heavens,  we  should  continue  to  see  it  for 
more  than  three  years  after  its  destruction ;  yet  Alpha  Centauri 
is  probably  our  nearest  neighbor  among  the  fixed  stars. 

553.  How  differences  of  parallax  may  be  detected. — Since  the  best 
astronomical  observations  are  liable  to  minute  errors,  which  ren- 
der it  difficult  to  determine  a  star's  absolute  place  with  the  accu- 
racy required  for  the  measurement  of  parallax,  astronomers  have 
sought  for  some  method  of  detecting  parallax  which  shall  be  free 
from  the  errors  of  ordinary  observations.     The  following  method 
has  been  proposed  for  this  purpose.  s"  rig.  iC5. 

Let  S  be  a  star  which  we  will  sup- 
pose to  have  a  visible  parallax,  and 
let  ABCD  be  the  small  ellipse  which 
it  appears  to  describe  in  consequence 
of  the  motion  of  the  earth  about  the 
sun.  Let  S'  and  S"  be  two  other 


294  ASTRONOMY. 

stars,  so  distant  from  the  earth  as  to  have  no  sensible  parallax, 
and  situated  on  the  axes  of  the  ellipse  ABCD,  and  suppose  the 
three  stars  to  be  included  in  the  same  field  of  the  telescope.  The 
apparent  distance  of  the  star  S  from  S'  will  change  during  the 
year  from  AS'  to  CS',  and  its  distance  from  S"  will  change  from 
BS"  to  DS".  These  distances  can  be  measured  with  great  exact- 
ness by  means  of  a  micrometer ;  and  these  measurements  are  in- 
dependent of  the  errors  which  affect  the  determination  of  the  ab- 
solute places  of  the  stars.  If  the  stars  S'  and  S"  have  a  small 
parallax,  then  these  observations  will  determine  the  difference  of 
parallax  between  the  star  S  and  the  stars  of  comparison.  When 
we  wish  to  select  stars  which  have  no  appreciable  parallax,  we 
choose  those  of  the  smallest  magnitude,  which  for  that  reason  are' 
presumed  to  be  at  the  greatest  distance  from  the  earth. 

55-1.  Parallax  of  61  Cygni. — By  the  method  here  indicated,  the 
parallax  of  the  star  61  Cygni  was  determined  by  the  great  as- 
tronomer Bessel,  of  Konigsberg,  to  be  0".35.  The  observations 
of  Mr.  Johnson,  at  Oxford,  make  the  parallax  of  this  star  0".4:0 ; 
Struve,  at  Pulkova,  makes  the  parallax  0".ol ;  and  Auwers,  at 
Konigsberg,  makes  the  parallax  0".56.  The  mean  of  these  four 
determinations  is  0".45,  indicating  a  distance  of  4i  millions  of 
millions  of  miles,  a  space  which  light  would  not  traverse  in  less 
than  7£  years. 

555.  Parallax  of  other  stars. — No  other  star  has  yet  been  found 
whose  parallax  exceeds  half  of  a  second.  Capella  has  a  parallax 
of  about  a  third  of  a  second,  and  j3  Centauri  nearly  half  a  second, 
and  there  are  six  smaller  stars  whose  parallax  is  not  less  than 
about  a  quarter  of  a  second.  All  the  other  stars  of  our  firma- 
ment are  apparently  at  a  greater  distance  from  us ;  and  if  the  dis- 
tance of  the  nearest  stars  is  so  great,  we  must  conclude  that  those 
faint  stars  which  are  barely  discernible  in  powerful  telescopes  are 
much  more  distant.  Ilence  we  conclude  that  we  do  not  see  them 
as  they  now  are,  but  as  they  were  years  ago ;  perhaps,  in  some 
instances,  with  the  rays  which  proceeded  from  them  several  thou- 
sands of  years  ago ;  and  it  is  possible  that  they  may  have  changed 
their  appearance,  or  have  been  entirely  annihilated  years  ago,  al- 
though we  actually  see  them  at  the  present  moment. 


THE  FIXED  STARS.  295 

556.  Light  of  the  sun  compared  with  that  of  the  fixed  stars. — The 
fixed  stars  must  be  self-luminous,  for  no  light  reflected  from  our 
sun  could  render  them  visible  at  the  enormous  distances  at  which 
they  are  situated  from  us.     Indeed,  it  is  demonstrable  that  many 
of  the  fixed  stars  actually  give  out  as  much  light  as  our  sun.     It 
is  estimated  that  the  light  of  our  sun  is  450,000  times  greater  than 
that  of  the  full  moon  ;  and  it  has  been  proved  that  the  light  of 
the  full  moon  is  13,000  times  greater  than  that  of  Sirius;  that  is, 
the  light  of  the  sun  is  about  6000  million  times  greater  than  that 
of  Sirius.     Since  the  quantity  of  light  which  the  eye  receives 
from  a  star  varies  inversely  as  the  square  of  its  distance,  and  since 
the  distance  of  Sirius  is  800,000  times  that  of  the  sun,  it  follows 
that,  if  Sirius  were  brought  as  near  to  us  as  the  sun,  its  light 
would  be  640,000  million  times  as  great  as  it  appears  at  present ; 
that  is,  the  light  emitted  by  Sirius  is  a  hundred  times  that  of  our 
sun.     Many  other  fixed  stars  probably  emit  as  much  light  as 
Sirius;  in  other  words,  the  fixed  stars  belong  to  the  same  class 
of  bodies  as  our  sun,  in  respect  of  the  amount  of  light  which  they 
emit ;  and  it  is  probable  that  many  of  them  are  bodies  of  at  least 
equal  dimensions,  otherwise  the  intensity  of  their  illumination 
must  be  very  much  greater  than  that  of  our  sun. 

557.  Proper  motion  of  the  stars. — The  changes  in  the  position  of 
the  stars  due  to  aberration  and  nutation  are  merely  apparent 
movements,  and  their  exact  amounts  can  be  readily  calculated  for 
any  star.    The  effects  of  precession  can  be  determined  with  equal 
facility.     It  is  found  by  observation,  however,  that  most  stars  ex- 
hibit a  slow  motion  in  the  heavens  which  can  not  be  thus  ac- 
counted for.     After  due  allowance  has  been  made  for  preces- 
sion, aberration,  and  nutation,  there  still  remain  very  appreciable 
changes  of  position.     These  are  not  such  periodical  motions  to 
and  fro  as  would  be  produced  by  parallax ;  on  the  contrary,  they 
are  uniformly  progressive  from  year  to  year.     A  star  in  Ursa 
Major  (known  as  1830  of  Groombridge's  catalogue)  travels  at  the 
rate  of  seven  seconds  in  a  year;  61  Cygni,  whose  parallax,  as  al- 
ready mentioned,  has  been  determined,  is  moving  at  the  rate  of 
five  seconds  annually.     The  star  Alpha  Centauri  has  a  proper 
motion  of  nearly  four  seconds  annually,  and  most  of  the  brighter 
stars  of  the  firmament  have  a  sensible  proper  motion.     The  result 
of  this  motion  is  a  slow  but  constant  change  in  the  figure  of  the 


296  ASTRONOMY. 

constellations.  In  the  case  of  several  of  the  stars,  this  change  in 
2000  years  has  become  quite  sensible  to  the  naked  eye.  The 
proper  motion  of  Arcturus  in  2000  years  has  amounted  to  more 
than  one  degree ;  that  of  Sirius  and  Procyon  to  three  quarters  of 
a  degree. 

558.  Cause  of  this  proper  motion. — There  are  two  ways  in  which 
such  movements  may  be  explained.     Either  the  star  itself  may 
be  supposed  to  have  a  real  motion  through  space,  or  the  sun,  at- 
tended by  the  planets,  may  have  a  real  motion  in  a  contrary  di- 
rection to  that  of  the  star's  apparent  one.     On  extending  the  in- 
quiry to  a  great  number  of  stars,  it  appears  beyond  doubt  that 
both  causes  must  be  in  existence,  certain  stars  having  really  an 
independent  motion  in  the  heavens,  which,  to  distinguish  it  from 
merely  apparent  displacements,  is  termed  improper  motion,  while 
the  solar  system  itself  travels  through  space. 

559.  How  could  a  motion  of  the  solar  system  be  detected  ? — If  we 
suppose  the  sun,  attended  by  the  planets,  to  be  moving  through 
space,  we  ought  to  be  able  to  detect  this  motion  by  an  apparent 
motion  of  the  stars  in  a  contrary  direction,  as,  when  an  observer 
moves  through  a  forest  of  trees,  his  own  motion  imparts  an  ap- 
parent motion  to  the  trees  in  a  contrary  direction.     All  the  stars 
would  not  be  equally  affected  by  such  a  motion  of  the  solar  sys- 
tem.    The  nearest  stars  would  appear  to  have  the  greatest  mo- 
tion, but  all  the  changes  of  position  would  appear  to  take  place  in 
the  same  direction.     The  stars  would  all  appear  to  recede  from 

Fig.  126.  Fig.  127. 


^**         rx  '•">  *' 

*-  »  A  •  —  *  —  »  — 


/  *  \  \  ^ 

/       I       \  *  * 

\  * 

A,  that  point  of  the  heavens  toward  which  the  sun  is  moving, 
while  in  the  opposite  quarter,  B,  the  stars  would  become  crowded 
more  closely  together. 


DOUBLE  STARS.  297 

560.  Direction  of  the  sun's  motion. — In  1783,  Sir  "William  Her- 
schel  announced  that  the  proper  motion  of  a  large  portion  of  the. 
stars  might  be  explained  by  supposing  that  the  sun  has  a  motion 
toward  a  point  in  the  constellation  Hercules.     More  recent  and 
extensive  investigations  have  not  only  established  the  fact  of  the 
solar  motion,  but  likewise  indicated  a  direction  very  nearly  coin- 
cident with  that  assigned  by  Herschel,  viz.,  nearly  toward  the  star 
p  Herculis.     The  average  displacement  of  the  stars,  as  estimated 
by  Struve,  indicates  that  the  motion  of  the  sun  in  one  year  is  about 
150  millions  of  miles,  which  is  about  one  fourth  of  the  velocity 
of  the  earth  in  its  orbit,  or  five  miles  per  second ;  but,  according 
to  the  estimate  of  Airy,  the  motion  of  our  solar  system  is  about 
twenty -seven  miles  per  second. 

561.  Is  the  sun's  motion  rectilinear? — It  is  probable  that  the  solar 
system  does  not  advance  from  age  to  age  in  a  straight  line,  but 
that  it  revolves  about  the  centre  of  gravity  of  the  group  of  stars 
of  which  it  forms  a  member.     It  is  also  probable  that  this  centre 
of  gravity  is  situated  nearly  in  the  plane  of  the  Milky  Way ;  and 
if  the  orbit  of  the  sun  is  nearly  circular,  this  centre  must  be  about 
90°  distant  from  p  Herculis,  the  point  toward  which  the  solar  sys- 
tem is  moving.     Maedler  conjectured  that  the  brightest  star  in 
the  Pleiades  was  the  central  sun  of  the  universe,  but  without  suf- 
ficient reason.     The  orbit  of  the  solar  system  is  probably  so  large 
that  ages  may  elapse  before  it  will  be  possible  to  detect  any  change 
in  the  direction  of  the  sun's  motion. 


CHAPTEE  XX. 

DOUBLE   STARS. — CLUSTERS   OF   STARS. — NEBULJE. 

562.  Double  stars.—  Many  stars  which  to  the  naked  eye,  or  with 
telescopes  of  small  power,  appear  to  be  single,  when  examined 
with  telescopes  of  greater  power  are  found  to  consist  of  two  stars 
placed  close  together.  These  are  called  double  stars.  Some  of 
these  are  resolved  into  separate  stars  by  a  telescope  of  moderate 
power,  as  Castor,  which  consists  of  two  stars  at  the  distance  of  5" 
from  each  other,  each  being  of  the  third  or  fourth  magnitude. 
Many  of  them,  however,  for  their  separation,  require  the  most 


298  ASTRONOMY, 

powerful  telescope.  Some  stars,  which  to  ordinary  telescopes  ap- 
pear only  double,  when  seen  through  more  powerful  instruments 
are  found  to  consist  of  three  stars,  forming  a  triple  star ;  and  there 
are  also  combinations  of  four,  five,  or  more  stars,  lying  within 
small  distances  from  each  other,  thus  forming  quadruple,  quintuple, 
and  multiple  stars.  Only  four  double  stars  were  known  until  the 
time  of  Sir  W.  Herschel,  who  discovered  upward  of  500,  and  sub- 
sequent observers  have  extended  this  number  to  6000. 

563.  Classification  of  double  stars.  —  Herschel  divided  double 
stars  into  four  classes,  according  to  the  angular  distance  between 
the  two  components.     The  first  class  comprised  those  only  in 
which  the  distance  between  the  two  components  does  not  exceed 
4" ;  the  second  class  those  in  which  it  exceeds  4",  but  falls  short 
of  8";  the  third  class  extends  from  8"  to  16";  and  the  fourth 
class  extends  from  16"  to  32".     Struve  has  subdivided  some  of 
Herschel's  classes,  making  thus   eight  classes   instead  of  four. 
When  the  distance  between  two  stars  exceeds  32",  they  are  not 
generally  admitted  into  the  catalogue  of  double  stars. 

In  some  instances  the  components  of  a  double  star  are  of  equal 
brilliancy,  but  it  more  frequently  happens  that  one  star  is  bright- 
er than  the  other.  Occasionally  the  inequality  of  light  is  so  great 
that  the  smaller  star  is  almost  lost  in  the  refulgence  of  its  bright- 
er neighbor. 

564.  Colored  stars. — Many  stars  shine  with  a  colored  light,  as 
red,  blue,  green,  or  yellow.     These  colors  are  exhibited  in  strik- 
ing contrast  in  many  of  the  double  stars.     Combinations  of  blue 
and  yellow,  or  green  and  yellow,  are  not  infrequent,  while  in 
fewer  cases  we  find  one  star  white  and  the  other  purple,  or  one 
white  and  the  other  red.     In  several  instances  each  star  has  a 
rosy  light     The  colors  of  the  two  components  are  sometimes 
complementary  to  each  other — that  is,  if  combined,  they  would 
form  white  light.     In  such  cases,  if  one  star  is  much  smaller  than 
the  other,  we  may  attribute  the  difference  of  color  to  the  effect  of 
contrast  only.     Thus,  if  the  larger  one  be  yellow,  the  companion 
may  incline  to  blue;  or,  if  the  former  have  a  greenish  light,  the 
latter  may  be  tinged  with  crimson.     Yet  it  can  hardly  be  doubt- 
ed that  in  many  cases  the  light  of  the  stars  is  actually  of  different 
colors ;  that  there  exist  in  the  universe  numbers  of  yellow,  blue, 


DOUBLE   STARS.  299 

green,  and  crimson  suns,  whose  refulgence  must  produce  the  most 
beautiful  effects  upon  the  planets  which  circulate  around  them. 

Single  stars  of  a  fiery  red  or  deep  orange  color  are  not  uncom- 
mon, but  there  is  no  instance  of  an  isolated  deep  blue  or  green 
star;  these  colors  are  apparently  confined  to  the  compound  stars. 

Below  the  constellation  Orion  there  is  a  star  of  the  seventh 
magnitude  of  a  blood-red  color,  and  near  it  is  another  star  of  sim- 
ilar brightness,  but  presenting  a  pure  white  light. 

The  following  are  a  few  of  the  most  interesting  colored  double 
stars : 

Name  of  star.  Color  of  Inrger  one.  Color  of  smaller  one. 

•y  Andromedts Orange Sea-green. 

a  Piscium Pale  green  -  -  -  Blue. 

/3  Cygni Yellow Sapphire-blue. 

a  Cassiopece Greenish  -  -  -  -  Fine  blue. 

A  star  in  Argo    -  -  -  -  Pale  rose  -  -  -  -  Greenish-blue. 
A  star  in  Centaurus  -  -  Scarlet Scarlet. 

565.  Stars  optically  double. — If  two  stars  be  very  nearly  in  the 
same  lino  of  vision,  though  one  may  be  vastly  more  distant  than 
the  other,  they  will  form  a  star  optically  double,  or  one  whose 
components  are  only  apparently  connected  by  the  near  coinci- 
dence of  their  directions  as  viewed  from  the  earth.  Thus  the 
two  stars  A  and  B,  seen  from 
the  earth  at  E,  will  appear  in 
close  juxtaposition,  although 
they  may  be  separated  by  an  interval  greater  than  the  distance 
of  the  nearest  from  the  earth.  The  chances,  however,  are  greatly 
against  there  being  a  large  number  of  stars  thus  optically  joined 
together.  If  the  stars  down  to  the  seventh  magnitude  were  scat- 
tered fortuitously  over  the  entire  firmament,  the  chances  against 
any  two  of  them  having  a  position  so  close  to  each  other  as  4" 
would  be  9000  to  1.  But  more  than  100  such  cases  of  juxtapo- 
sition are  known  to  exist. 

560.  Binary  stars.— In  the  year  1780,  Sir  William  Ilerschel  un- 
dertook an  extensive  series  of  observations  of  double  stars,  record- 
ing the  relative  position  of  the  components,  and  the  distance  by 
which  they  were  separated.  By  this  means  he  hoped  to  be  able 
to  detect  a  parallax.  lie  found  that  the  distance  and  relative  po- 


300 


ASTRONOMY. 


sition  of  the  components  of  a  double  star  were  subject  to  change, 
but  the  period  of  this  change  had  no  relation  to  the  earth's  mo- 
tion about  the  sun.  After  twenty  years  of  observations,  he  ascer- 
tained and  announced  that  these  apparent  changes  of  position 
were  due  to  real  motions  of  the  stars  themselves;  that  the  com- 
ponents of  several  of  the  double  stars  moved  in  orbits  in  the  same 
manner  as  the  planets  move  around  the  sun ;  that  there  exist  si- 
dereal systems  consisting  of  two  stars  revolving  about  each  other, 
or,  rather,  both  revolving  round  their  common  centre  of  gravity. 
These  stars  are  termed  physically  double,  or  binary  stars,  to  dis- 
tinguish them  from  other  double  stars  in  which  no  such  periodic 
change  of  position  has  been  discovered. 

567.  The  star  Gamma  Virginia. — One  of  the  most  remarkable 
of  the  binary  stars  is  j  Virginia  This  is  a  star  of  the  fourth 
magnitude,  and  its  components  are  almost  exactly  equal.  It  has 
been  known  to  consist  of  two  stars  since  1718,  their  distance  be- 
ing then  7";  and  since  1780  they  have  been  regularly  observed. 
In  1836,  their  distance  from  each  other 
was  less  than  half  a  second,  so  that  no  tel- 
escope, unless  of  a  very  superior  quality, 
could  show  them  otherwise  than  as  a  sin- 
gle star.  At  present  their  distance  from 
each  other  is  about  4".  During  the  inter- 
val of  146  years,  the  direction  of  one  of  the 
components,  as  seen  from  the  other,  has 
changed  by  nearly  360°.  The  entire  se- 
ries of  observations  is  well  represented 
by  supposing  each  of  the  stars  to  revolve 
about  their  common  centre  of  gravity  in 
an  ellipse  whose  major  axis  is  7",  and  in  a 
period  of  169  years.  In  the  annexed  fig- 
ure, the  dotted  line  represents  the  appar- 
ent orbit  of  one  of  the  stars  about  the  oth- 
er, while  the  black  line  represents  the  form 
of  the  actual  orbit  as  computed.  The  orb- 
it, as  viewed  from  the  earth,  is  seen  somewhat  obliquely,  and  the 
apparent  length  of  the  mnjor  axis  is  thereby  somewhat  reduced. 


Fig.  129. 


568.  The  star  70  Ojrftiuchi. — The  star  70  Ophiuchi  consists  of 


DOUBLE  STARS. 


801 


two  components,  one  of  the  fourth,  the  rig.  iso.  s 
other  of  the  seventh  magnitude.  Since 
1779,  one  has  made  nearly  a  complete 
revolution  about  the  other  in  an  el- 
lipse whose  major  axis  is  about  8",  and 
the  period  is  about  92  years.  In  the 
annexed  figure,  the  dotted  line  repre- 
sents the  apparent  orbit  of  one  of  the 
stars  about  the  other,  while  the  black 
line  represents  the  form  of  the  actual 
orbit  as  computed. 

569.  The  star  Xi  Ursce,  Majoris,  etc. — The  star  £  Urste  Majoris 
consists  of  two  components,  one  of 

the  fourth,  the  other  of  the  fifth  mag- 
nitude. Since  1780,  one  has  com- 
pleted an  entire  revolution  about  the 
other,  and  has  entered  upon  a  second 
period.  The  major  axis  of  the  orbit 
is  about  5",  and  the  time  of  revolu- 
tion 61  years.  The  annexed  figure 
represents  both  the  apparent  and  the 
real  orbit. 

The  star  %  Herculis  consists  of  two 
components,  one  of  the  third,  the  other  of  the  sixth  magnitude. 
Since  1782,  one  has  completed  two  .entire  revolutions  about  the 
other  in  an  ellipse  whose  major  axis  is  2£",  and  the  period  of  a 
revolution  is  36  years. 

570.  The  star  Alpha  Ccntauri.— The  star  a  Centauri  consists  of 
two  components,  one  of  the  first,  the  other  of  the  second  magni- 
tude.    These  two  stars  were  observed  by  Lacaille  in  1751 ;  and 
since  1826  their  positions  have  been  frequently  and  carefully  ob- 
served.    The  annexed  figure  represents  the  apparent  path  from 


302  ASTRONOMY. 

1826  to  1864,  as  well  as  the  remaining  portion  of  the  probable 
apparent  orbit.  The  major  axis  of  this  orbit  is  about  30'',  and 
the  time  of  one  revolution  about  80  years. 

571.  Number  of  the  binary  stars. — There  are  358  double  stars 
whose  physical  connection  is  believed  to  be  established  beyond 
doubt.     Of  these,  13  have  completed  one  or  more  revolutions 
since  they  were  first  observed;  28  have  completed  more  than  a 
half  revolution ;  43  have  completed  more  than  a  quarter  revolu- 
tion; and  156  have  advanced  more  than  twenty  degrees  in  their 
orbits.    There  are  558  double  stars  whose  physical  connection  is 
believed  to  be  highly  probable,  if  not  entirely  certain.    Of  these, 
there  are  22  whose  periods  are  less  than  a  century,  and  there  are 
36  whose  periods  are  less  than  two  centuries;  but  the  periods  of 
most  of  the  binary  stars  apparently  exceed  a  thousand  years. 

572.  The  law  of  gravitation  extends  to  the  fixed  stars. — It  has  been 
proved,  in  Art.  249,  that  if  a  body  revolve  in  an  ellipse  by  an  at- 
tractive force  directed  toward  the  focus,  that  force  must  vary  in- 
versely as  the  square  of  the  distance.     But  several  of  the  binary 
stars  have  been  proved  to  revolve  in  ellipses ;  hence  it  is  inferred 
that  the  same  law  of  gravitation  which  prevails  in  the  solar  sys- 
tem prevails"  among  the  sidereal  systems. 

573.  Absolute  dimensions  of  the  orbit  of  a  Unary  star. — If  we  knew 
the  distance  of  a  binary  star  from  the  earth,  we  could  compute 
the  absolute  dimensions  of  the  orbit  described.     Now  a  Centauri 
and  61  Cygni  are  both  binary  stars,  and  their  distances  are  toler- 
ably well  determined.     The  distance  of  a  Centauri  is  224,000 
times  the  radius  of  the  earth's  orbit.     Hence  we  shall  have  the 
proportion 

E:  224,000::  tang.  15":  16; 

that  is,  the  radius  of  the  orbit  described  by  the  components  of 
a  Centauri  is  16  times  the  radius  of  the  earth's  orbit,  or  about 
four  fifths  the  distance  of  Uranus  from  the  sun.  In  a  similar 
manner  it  has  been  computed  that  the  radius  of  the  orbit  de- 
scribed by  the  components  of  61  Cygni  is  34  times  the  radius  of 
the  earth's  orbit,  which  is  considerably  greater  than  the  orbit  of 
Neptune. 


DOUBLE   STARS,  ETC.  303 

574.  Mass  of  a  Unary  star  computed. — Since  the  relation  be- 
tween the  dimensions  of  the  orbit  and  the  time  of  revolution  de- 
termines the  relative  masses  of  the  central  bodies,  we  are  enabled 
to  compare  the  mass  of  a  binary  star  with  that  of  our  sun,  when 
we  know  the  absolute  radius  of  the  orbit,  and  the  periodic  time 
of  the  star.  In  Art.  469  we  found 

If  TO  represent  the  mass  of  our  sun,  r  the  radius  of  the  earth's 
orbit,  and  t  the  time  of  the  earth's  revolution,  or  one  year,  then 

M  =  — .TO. 

In  the  case  of  a  Centauri,  K  has  been  found  equal  to  16,  and 
T= 80  years.  Hence 


that  is,  the  mass  of  the  double  star  a  Centauri  is  about  three  fifths 
that  of  our  sun.  In  a  similar  manner  it  has  been  computed  that 
each  of  the  stars  which  compose  61  Cygni  is  about  one  third  of 
our  sun. 

In  1862  it  was  discovered  that  Sir! us,  the  brightest  star  in  the 
heavens,  is  attended  by  a  minute  companion  of  the  ninth  magni- 
tude, and  it  has  since  been  proved  that  the  two  stars  are  physi- 
cally connected.  It  lias  been  computed  that  the  time  of  one  rev- 
olution is  49  years;  that  the  mass  of  Sirius  is  13.7  times  that  of 
our  sun,  and  the  mass  of  its  companion  6.7  times  that  of  our  sun. 
The  mass  of  70  Ophiuchi  and  its  companion  has  been  computed 
to  be  three  times  that  of  our  sun. 

575.  The  fixed  stars  are  suns. — "We  thus  see  that  the  stars  are  bod- 
ies essentially  like  our  sun.    Some  of  them  have  a  power  of  attrac- 
tion nearly  equal  to  that  of  our  sun,  and  it  is  probable  that  others 
have  a  greater  power  of  attraction.     Some  of  them  emit  more 
light  than  our  sun.    The  stars  are  therefore  self-luminous  bodies 
of  vast  size,  and  are  entitled  to  be  called  suns.   In  the  binary  stars, 
then,  we  have  examples,  not  of  planets  revolving  round  a  sun,  as 
in  our  solar  system,  but  of  sun  revolving  round  sun. 

576.  Triple  stars. — Besides  the  binary  stars,  there  are  some  triple 
stars  which  are  proved  to  be  physically  connected.    Of  these  the 


304 


ASTRONOMY. 


most  remarkable  is  Zeta  Cancri.  It  consists  of  three  components, 
one  being  of  the  sixth,  and  the  other  two 
of  the  ninth  magnitude.  Since  1781,  one 
of  these  components  has  made  a  complete 
revolution  about  one  of  the  others  in  an 
ellipse  whose  major  axis  is  2",  and  the  pe- 
riod of  revolution  is  58  years.  During  the 
same  period  the  other  component  has  ad- 
vanced more  than  30  degrees  in  its  orbit, 
from  which  it  is  estimated  that  its  period 
of  revolution  must  be  about  500  years.  The 
annexed  diagram  represents  the  orbit  of  the 
nearest  component,  and  a  portion  of  the  or- 
bit of  the  more  remote  component. 

The  star  51  Libra)  is  a  triple  star,  two  of 
whose  components  are  of  the  fifth  magni- 
tude, and  the  other  of  the  seventh.  It  has 
been  computed  that  the  nearer  component  makes  a  revolution  in 
105  years,  and  the  more  remote  component  in  600  or  700  years. 

\  577.  Quadruple  and  quintuple  stars. — £  Lyra3  furnishes  an  in- 
stance of  a  quadruple  star,  in  which  all  the  components  are  be- 
lieved to  be  physically  connected.  Three  of  the  components  are 
of  the  fifth,  and  the  other  of  the  sixth  magnitude.  The  move- 
ment of  these  stars  is  extremely  slow,  and,  at  the  present  rate  of 
motion,  it  will  require  nearly  a  thousand  years  for  the  nearest 
component  to  complete  one  revolution,  and  many  thousand  years 
for  the  most  remote  component. 

Theta  Orionis  is  a  quintuple  star,  in  which  one  of  the  com- 
ponents is  of  the  sixth  magnitude,  two  are  of  the  seventh,  one  of 
the  eighth,  and  the  other  is  of  the  fifteenth  magnitude.  In  the 
best  telescopes,  a  sixth  star  of  extreme  faintness  may  also  be  seen. 
The  relative  position  of  these  stars  has  not  sensibly  changed  since 
they  were  first  observed  by  Herschel,  and  it  is  uncertain  whether 
they  are  physically  connected. 

578.  Clusters  of  stars. — In  many  parts  of  the  heavens  we  find 
stars  crowded  together  in  clusters,  frequently  in  such  numbers  as 
to  defy  all  attempts  to  count  them.  Some  of  these  clusters  are 
visible  to  the  naked  eye.  In  the  cluster  called  the  Pleiades, 


CLUSTERS  OF  STARS. — NEBULAE.  305 

six  stars  are  readily  perceived  by  the  naked  eye,  and  we  obtain 
glimpses  of  many  more.  With  a  telescope  of  moderate  power 
188  stars  can  be  counted. 

In  the  constellation  Cancer  is  a  luminous  spot  called  Preesepe, 
or  the  bee-hive,  which  a  telescope  of  moderate  power  resolves  en- 
tirely into  stars.  There  is  a  remarkable  group  in  the  sword-han- 
dle of  Perseus,  in  which  the  stars  are  readily  seen  with  a  common 
night-glass,  though  the  whole  have  a  blurred  aspect  to  the  naked 
eye. 

One  of  the  most  magnificent  clusters  in  the  northern  hemi- 
sphere occurs  in  the  constellation  Hercules,  between  the  stars  r\ 
and  £.  It  is  visible  to  the  naked  eye  on  clear  nights  as  a  hazy- 
looking  object,  and  the  stars  composing  it  are  readily  seen  with  a 
telescope  of  moderate  power.  When  examined  in  a  powerful  in- 
strument, its  aspect  is  grand  beyond  description ;  the  stars,  which 
are  coarsely  scattered  at  the  borders,  come  up  to  a  perfect  blaze 
in  the  centre. 

The  richest  cluster  in  the  entire  heavens  is  situated  in  the  con- 
stellation Centaurus,  which  belongs  to  the  southern  hemisphere, 
and  is  called  w  Centauri.  To  the  naked  eye  it  appears  like  a  neb- 
ulous or  hazy  star  of  the  fourth  magnitude,  while  in  the  telescope 
it  is  found  to  cover  a  space  two  thirds  of  the  apparent  diameter 
of  the  moon,  over  which  the  stars  are  congregated  in  countless 
numbers.  See  Plate  VIL,  Fig.  2. 

We  can  not  doubt  that  most  of  the  stars  in  such  a  cluster  as 
to  Centauri  are  near  enough  to  each  other  to  feel  each  other's  at- 
traction. They  must  therefore  be  in  motion,  and  we  must  regard 
this  cluster  as  a  magnificent  astral  system,  consisting  of  a  count- 
less number  of  suns,  each  revolving  in  an  orbit  about  the  common 
centre  of  gravity. 

579.  Nebulce.. — In  various  parts  of  the  firmament  we  discover 
with  a  telescope  dim  patches  of  light,  presenting  a  hazy,  unde- 
fined, or  cloud-like  appearance.  These  objects  are  called  nebulce. 
A  large  proportion  are  either  round  or  oval,  brighter  toward  their 
centres  than  at  their  borders,  and  when  viewed  with  small  optical 
power  very  much  resemble  comets,  for  which  they  are  often  mis- 
taken. In  more  powerful  instruments,  such  as  those  brought  into 
use  by  Sir  William  Herschel,  a  considerable  number  are  readily 
resolved  into  clusters  of  stars,  like  Prasepe,  or  the  group  in  Per- 
il 


306  ASTRONOMY. 

sens  above  mentioned ;  some  hundreds,  or  even  thousands  of  stars 
are  wedged  together  within  the  space  of  two  or  three  minutes  of 
arc  or  less.  Many  others  present  a  mottled,  glittering  aspect 
when  thus  viewed,  which  shows  that  they  are  similarly  consti- 
tuted, but  too  distant  for  our  telescopes  to  separate  them  into 
stars ;  while,  as  might  be  expected,  there  are  also  very  many  that 
the  most  powerful  optical  means  hitherto  devised  have  altogether 
failed  to  exhibit  otherwise  than  as  faint,  cloud-like  objects.  Very 
many  of  them  have  the  same  form  and  general  appearance  as  the 
resolvable  nebulae  seen  in  common  telescopes,  and  hence  there  is 
reason  for  supposing  them  to  be  similar  clusters  of  stars,  but  situ- 
ated at  far  greater  distances  from  the  earth.  About  5000  nebulae 
have  been  observed,  and  their  places  are  assigned  in  catalogues. 
The  following  are  among  the  most  remarkable  of  this  class  of 
objects. 

580.  The  great  nebula  in  Andromeda. — This  is  a  very  conspicu- 
ous nebula,  distinctly  visible  without  a  telescope,  and  is  often  mis- 
taken for  a  comet.     It  was  discovered  nearly  1000  years  ago, 
though  not  much  noticed  until  attention  was  directed  to  its  sin- 
gular appearance  by  Simon  Marius  in  1612.     It  is  of  an  oval 
shape,  and  has  been  described  as  resembling  the  light  of  a  candle 
shining  through  horn.     When  observed  with  the  best  telescopes, 
its  boundaries  appear  greatly  extended,  its  extreme  length  being 
90  minutes,  and  its  breadth  15  minutes.     The  great  telescope  at 
Cambridge  Observatory  exhibits  two  dark  bands  or  canals  nearly 
straight  and  parallel,  about  one  degree  in  length,  running  in  the 
direction  of  the  longer  axis  of  the  nebula.     Till  very  recently, 
this  nebula  defied  all  the  optical  power  that  could  be  brought  to 
bear  upon  it  to  resolve  it  into  stars,  or  even  to  afford  any  symp- 
toms of  its  stellar  character.     But  within  the  last  few  years,  de- 
cisive evidence  of  its  consisting  of  stars  has  been  obtained  with 
the  Cambridge  telescope.     Plate  VII.,  Fig.  1,  gives  a  representa- 
tion of  the  appearance  of  this  nebula  in  Herschel's  telescope. 

581.  The  great  nebula  in  the  sword-handle  of  Orion. — This  nebula 
was  first  discovered  and  figured  by  Huygens  in  1659.     It  consists 
of  irregular  nebulous  patches,  extending  over  a  surface  about  40 
minutes  square,  its  superficial  magnitude  being  more  than  twice 
that  of  the  moon's  disc.     The  brightest  portion  of  the  nebula  re- 


NEBULA.  307 

sembles  the  head  and  jaws  of  some  monstrous  animal  with  an 
enormous  proboscis.  The  nebulosity  in  this  vicinity  is  flocculent, 
and  of  a  greenish- white  tinge.  It  was  irresolvable  until  the  com- 
pletion of  Lord  Eosse's  telescope ;  but  in  this  instrument  there 
are  strong  indications  of  its  being  composed  of  a  vast  multitude 
of  stars,  far  removed  from  us  in  the  profundity  of  space.  A  com- 
parison of  the  earlier  with  the  more  recent  representations  of  this 
nebula  might  lead  to  the  conclusion  that  it  had  changed  its  form 
within  two  hundred  years,  but  no  such  conclusion  can  be  safely 
drawn  on  account  of  the  imperfection  of  the  telescopes  with  which 
the  early  observations  were  made.  Plate  VII.,  Fig.  3,  gives  a  rep- 
resentation of  the  appearance  of  this  nebula. 

582.  The  spiral  nebula. — This  nebula  is  situated  near  the  ex- 
tremity of  the  tail  of  the  Great  Bear.    It  is  a  double  nebula,  with 
two  centres  about  five  minutes  apart.     From  one  of  the  centres 
proceed  several  luminous  streams,  which  wind  spirally  round  the 
nucleus,  suggesting  the  idea  of  a  body  not  in  a  state  of  permanent 
equilibrium.     Though  not  clearly  resolved  into  stars  with  Lord 
Eosse's  telescope,  some  evidence  is  thereby  afforded  that  it  is  so 
composed.     Other  nebulse  have  similar  spiral  coils,  but  less  dis- 
tinctly marked  than  in  the  one  above.     Plate  VIII.,  Fig.  4,  gives 
a  representation  of  this  nebula. 

583.  The  dumb-bell  nebula. — This  nebula  is  situated  between  the 
constellations  Swan  and  Eagle.     In  a  small  telescope  it  exhibits 
two  centres,  connected  by  a  nebulous  band,  its  entire  diameter  be- 
ing 7  or  8  minutes.     Sir  John  Herschel  compared  its  appearance 
to  that  of  a  dumb-bell.     In  Lord  Eosse's  telescope  the  form  ap- 
pears less  regular,  but  its  general  outline  is  elliptical.    Plate  VEIL, 
Fig.  1,  gives  a  representation  of  this  nebula. 

584.  The  crab  nebula.— Tins  nebula  is  situated  near  the  star  £ 
in  the  southern  horn  of  Taurus.     In  an  ordinary  telescope  it  ap- 
pears of  an  oval  form,  but  in  Lord  Eosse's  telescope  it  is  seen  as 
a  densely-crowded  cluster,  with  branches  streaming  off  from  the 
oval  boundary  like  claws,  so  as  to  give  it  an  appearance  that  in  a 
measure  justifies  the  name  of  the  crab  nebula  by  which  it  is  often 
distinguished.     Plate  VIIL,  Fig.  2,  gives  a  representation  of  this 
nebula. 


308  ASTRONOMY. 

585.  The  annular  nebula  in  Lyra. — This  nebula  is  situated  in 
the  constellation  Lyra,  between  the  stars  /3  and  -y.    In  Sir  J.  Her- 
schel's  telescope  it  appeared  like  a  ring  of  light  of  a  somewhat 
oval  form.     The  centre  was  not  entirely  black,  but  filled  with  a 
faint  nebulous  light.     In  Lord  Rosse's  telescope  are  seen  fringes 
extending  from  each  side  of  the  annulus,  and  also  stripes  crossing 
the  central  portion.    Though  apparently  a  small  nebula,  its  actual 
dimensions  must  be  enormous.     Even  supposing  it  no  farther 
from  us  than  61  Cygni,  the  diameter  of  the  ring  would  be  20,000 
millions  of  miles,  and  it  is  not  improbable  that  its  real  distance  is 
incomparably  greater  than  that  of  the  above  star.     Plate  VIII., 
Fig.  3,  gives  a  representation  of  this  nebula. 

586.  Planetary  nebula?. — Planetary  nebula)  exhibit  discs  of  uni- 
form brightness  throughout,  often  very  sharply  defined  at  the  bor- 
ders, or  only  a  little  curdled  or  furred,  as  the  edges  of  a  planet 
frequently  appear  when  the  night  is  unfavorable  for  telescopic 
observation.     They  are  called  planetary  nebulae  from  the  great 
resemblance  they  offer  to  the  discs  of  planets.     Not  far  from  the 
star  j3  in  Ursa  Major  is  a  fine  nebula  of  this  kind.     It  is  circular, 
nearly  3  minutes  in  diameter,  and  of  equable  light  on  its  whole 
surface,  and,  after  a  long  inspection,  looks  like  a  condensed  muss 
of  attenuated  light,  seemingly  of  the  size  of  Jupiter.     Supposing 
it  placed  at  a  distance  from  us  not  more  than  that  of  61  Cygni, 
it  would  have  a  linear  diameter  seven  times  greater  than  that  of 
the  orbit  of  Neptune.    About  twenty  planetary  nebulae  have  been 
observed.     They  can  not  be  globular  clusters  of  stars,  otherwise 
they  would  be  brighter  in  the  middle  than  at  the  borders.     It 
has  been  conjectured  that  they  may  be  hollow  spherical  shells,  or 
circular  flat  discs,  whose  planes  are  nearly  at  right  angles  to  our 
line  of  vision. 

587.  Nebulous  stars. — Nebulous  stars  are  stars  surrounded  by  a 
faint  nebulosity,  usually  of  a  circular  form,  and  sometimes  several 
minutes  in  diameter.    In  some  cases  the  nebulosity  is  sharply  de- 
fined at  the  borders,  in  others  it  gradually  fades  away  to  dark- 
ness.    The  stars  thus  attended  have  nothing  in  their  appearance 
to  distinguish  them  from  ordinary  stars,  nor  does  the  nebulosity 
in  which  they  are  situated  offer  the  slightest  indications  of  resolv- 
ability  into  stars  with  any  telescopes  hitherto  constructed.     As 


NEBULA.  309 

instances  of  nebulous  stars  may  be  mentioned  one  of  the  fifth  mag- 
nitude, numbered  55  in  Andromeda;  and  another  of  the  same 
brightness,  numbered  8  in  Canes  Venatici. 

588.  Distribution  of  the  nebulae. — The  nebulie  are  not  distributed 
uniformly  over  all  parts  of  the  heavens.     From  certain  regions 
they  are  wholly  absent ;  in  others  they  are  rarely  found ;  while 
in  other  regions  they  are  crowded  in  amazing  profusion.     They 
are  most  numerous  in  the  constellations  Leo,  Virgo,  and  Ursa 
Major. 

589.  Observations  with  the  spectroscope. — The  spectroscope  has 
furnished  important  information  respecting  the  condition  of  the 
fixed  stars  and  the  nebula3.     About  one  third  of  all  the  stars 
which  have  been  examined  exhibit  spectra  exactly  like  that  of 
our  sun,  showing  dark  lines  occupying  the  same  position  as  in  the 
solar  spectrum,  proving  that  they  are  composed  of  the  same  ele- 
ments as  our  sun,  and  are  in  the  same  physical  condition.     About 
one  half  of  all  the  stars  exhibit  the  colors  of  the  solar  spectrum 
crossed  by  four  very  heavy  black  lines  produced  by  the  presence 
of  hydrogen,  while  some  of  the  brighter  stars  exhibit  a  few  other 
dark  lines.     Nearly  one  sixth  of  all  the  stars  exhibit  a  spectrum 
showing  not  only  dark  lines,  but  a  number  of  bright  bands  sep- 
arated by  dark  bands ;  while  a  few  have  been  found  whose  spec- 
tra exhibit  not  only  dark  lines,  but  also  bright  lines  which  are  very 
brilliant,  like  those  belonging  to  the  spectrum  of  a  gaseous  body. 
These  bright  lines  indicate  the  presence  of  incandescent  gas. 

About  one  third  of  the  nebulas  which  have  been  examined  ex- 
hibit spectra  consisting  merely  of  a  few  isolated  bright  lines  which 
in  several  cases  correspond  to  well-known  gases.  Hence  it  is  in- 
ferred that  these  bodies  do  not  consist,  like  our  sun,  of  a  white-hot 
nucleus  enveloped  in  an  atmosphere  which  absorbs  a  portion  of 
the  rays,  but  that  they  are  in  the  condition  of  a  luminous  gas. 
About  two  thirds  of  all  the  nebula)  exhibit  a  spectrum  which  is 
apparently  continuous.  It  is  inferred  that  the  latter  consist  of  an 
assemblage  of  stars — that  is,  are  clusters  of  stars ;  while  the  former 
are  masses  of  glowing  gas — that  is,  are  not  capable  of  being  re- 
solved into  stars. 

These  facts  are  regarded  as  proving  the  existence  of  true  nebu- 
las ;  that  is,  of  vast  masses  in  the  gaseous  condition. 


310  ASTRONOMY. 

^590.  Have  any  nebula;  changed  their  forms? — The  forms  of  many 
of  the  nebulas  are  so  peculiar  that  it  is  difficult  to  regard  them  as 
having  attained  a  condition  of  permanent  equilibrium,  and  it  has 
been  supposed  that  we  see  them  now  in  the  state  of  transition  to- 
ward stable  forms.  A  comparison  of  the  present  appearance  of 
many  nebulas  with  the  representations  of  them  furnished  by  for- 
mer astronomers  would  lead  to  the  conclusion  that  they  had  sen- 
sibly changed  their  form  within  100  years.  Such  a  conclusion 
may  be  premature,  but  it  is  probable  that  future  astronomers  will 
discover  changes  that  are  incontestable.  If  any  of  the  nebulas 
consist  wholly  of  vaporous  matter,  they  are  probably  in  a  state 
of  gradual  condensation ;  and  if  they  all  consist  of  clusters  of  stars, 
then  these  stars  are  doubtless  in  motion,  forming  astral  systems 
of  wonderful  complexity. 

591.  Variations  in  the  brightness  of  nebula*. — Some  of  the  nebulaa 
have  exhibited  decided  changes  of  brightness.     A  nebula,  situ' 
ated  near  E  in  Taurus,  at  the  date  of  its  discovery  in  1852  was 
easily  seen  with  a  good  telescope,  whereas  in  1862  it  was  invisi- 
ble with  instruments  of  far  greater  power.     A  small  star  close  to 
this  nebula  likewise  faded  within  the  same  lapse  of  time.    Anoth- 
er nebula,  situated  near  the  Pleiades,  in  1859  could  be  seen  with 
a  three-inch  telescope,  whereas  in  1862  it  could  only  be  seen  with 
difficulty  through  the  largest  telescope.     Five  or  six  cases  of  this 
kind  have  been  noticed.     It  is  not  improbable  that  tbese  varia- 
tions of  brightness  are  due  to  the  same  cause  as  the  changes  of 
the  variable  stars. 

592.  The  Via  Lactea,  or  Milky  Way.  —  The  Galaxy,  or  Milky 
Way,  is  that  whitish  luminous  band  of  irregular  form  which  is 
seen  on  a  clear  night  stretching  across  the  expanse  of  heaven  from 
one  side  of  the  horizon  to  the  other.     To  the  naked  eye  it  pre- 
sents merely  a  diffused  milky  light,  stronger  in  some  parts  than 
in  others ;  but  when  examined  in  a  powerful  telescope,  it  is  found 
to  consist  of  myriads  of  stars  so  small  that  no  one  of  them  singly 
produces  a  sensible  impression  on  the  unassisted  eye. 

The  general  course  of  the  Milky  "Way  is  in  a  great  circle,  in- 
clined about  63°  to  the  celestial  equator,  and  intersecting  it  near 
the  constellations  Orion  and  Ophiuchus. 

The  distribution  of  the  telescopic  stars  within  its  limits  is  far 
from  uniform.  In  some  regions  several  thousands  are  crowded 


THE  MILKY  WAY.  311 

together  within  the  space  of  one  square  degree ;  in  others  only  a 
few  glittering  pointst  are  scattered  upon  the  black  ground  of  the 
heavens.  In  some  parts  it  presents  to  the  naked  eye  a  bright 
glow  of  light  from  the  closeness  'of  the  constituent  stars ;  in  oth- 
ers there  are  dark  spaces  containing  scarcely  a  single  star.  Such 
vacancies  occur  in  the  constellations  Scorpio  and  Ophiuchus. 

593.  Law  of  distribution  of  the  stars. — In  order  to  decide  whether 
the  stars  are  distributed  over  the  surface  of  the  heavens  accord- 
ing to  any  general  law,  Sir  W.  Herschel  undertook  a  rigorous  tel- 
escopic survey  of  the  heavens,  counting  the  number  of.  the  stars 
visible  in  the  field  of  his  telescope  when  directed  to  different  parts 
of  space.     He  thus  discovered  that  around  the  poles  of  the  Milky 
Way  the  stars  are  more  thinly  scattered  than  elsewhere ;  that  as 
we  advance  toward  the  Milky  Way  the  number  of  stars  included 
in  the  field  of  view  of  the  telescope  increases,  at  first  slowly,  but 
afterward  more  rapidly ;  and  that  along  the  Milky  Way  the  stars 
are  crowded  so  closely  together  that  it  becomes  in  many  cases  im- 
possible to  count  them. 

594.  Hypothesis  of  Sir  William  Herschel— In  1784,  Sir  W.  Her- 
schel  advanced  the  following  hypothesis  respecting  the  Milky 
Way :  The  stars  of  our  firmament,  instead  of  being  scattered  in 
all  directions  promiscuously  through  space,  constitute  a  cluster 
with  definite  limits,  in  the  form  of  a  stratum,  of  which  the  thick- 
ness is  small  in  comparison  with  its  length  and  breadth,  and  in 
which  the  earth  occupies  a  position  somewhere  about  the  middle 
of  its  thickness.     For  if  we  suppose  the  stars  to  be  scattered  pret- 
ty uniformly  through  space,  the  number  of  stars  visible  in  the 
field  of  a  telescope  ought  to  be  about  the  same  in  every  direction, 
provided  the  stars  extend  in  all  directions  to  an  equal  distance. 
But  if  the  stars  about  us  compose  a  stratum^  whose  thickness  is 
small  in  comparison  with  its  length  and  breadth,  then  the  num- 
ber of  stars  visible  in  the  different  directions  will  lead  us  to  a 
knowledge  both  of  the  exterior  form  of  this  stratum,  and  of  the 
place  occupied  by  the  observer.    For  example,  if  within  a  certain 
circle  of  the  heavens  we  count  ten  stars,  and  in  a  circle  of  the 
same  diameter,  taken  in  a  different  direction,  we  count  eighty  stars 
with  the  same  telescope,  the  lengths  of  the  two  visual  rays  will  be 
in  the  ratio  of  1  to  2,  or  the  cube  roots  of  1  and  8. 


312  ASTRONOMY. 

595.  This  hypothesis  is  untenable. — This  hypothesis  assumes,  1st, 
that  the  stars  are  uniformly  distributed  through  space ;  and,  2d, 
that  Herschel  was  able  with  his  telescope  to  penetrate  to  the  lim- 
its of  our  stratum. 

At  a  later  period  of  his  life,IIerschel  abandoned  each  of  these 
hypotheses.  Every  increase  in  the  power  of  his  telescopes  dis- 
closed new  stars  which  before  had  been  invisible,  and  he  was 
compelled  to  admit  that  with  his  telescope  of  40  feet  the  Milkv 
Way  was  entirely  fathomless  ;  and  instead  of  the  stars  being  dis- 
tributed uniformly  through  space,  he  admitted  that  there  is  a 
great  and  sudden  condensation  of  stars  in  the  neighborhood  of  the 
Milky  Way. 

In  every  part  of  the  heavens  the  stars  seem  to  extend  to  a  dis- 
tance beyond  the  reach  of  the  most  powerful  telescope  hitherto 
constructed,  and  hence  the  shape  of  that  portion  of  space  which 
the  stars  occupy  must  be  entirely  unknown  to  us ;  that  is,  the 
material  universe  appears  to  us  to  be  boundless. 

596.  Madler  s  hypotfiesis  respecting  the  Milky  Way. — Madler  sup- 
poses that  the  stars  of  the  Milky  Way  are  grouped  together  in 
the  form  of  an  immense  ring,  or  perhaps  a  system  of  detached  but 
concentric  star-rings  of  unequal  thickness  and  various  dimensions, 
but  all  situated  nearly  in  the  same  plane.    To  an  observer  situated 
in  the  centre  of  such  a  system  of  rings,the  inner  ring  would  seem  to 
cover  the  exterior  ones ;  that  is,  the  stars  would  seem  to  form  but 
a  single  ring,  and  this  ring  would  be  a  great  circle  of  the  sphere. 
The  Milky  Way,  in  fact,  divides  our  firmament  into  two  portions, 
whose  areas  are  to  each  other  in  the  ratio  of  about  8  to  9,  from 
which  it  is  concluded  that  the  solar  system  is  not  situated  exactly 
in  the  plane  of  the  Milky  Way,  but  somewhat  toward  the  south, 
or  in  the  direction  of  the  constellation  Virgo. 

The  division  of  tjie  Milky  Way  throughout  a  considerable  por- 
tion of  its  extent  into  two  separate  branches  indicates  that  in  this 
part  of  the  firmament  the  star-rings  do  not  cover  each  other,  which 
Madler  explains  by  supposing  that  we  are  eccentrically  situated, 
being  nearer  to  the  southern  than  to  the  northern  part  of  the 
rings.  This  supposition  would  also  explain  the  greater  brilliancy 
of  the  Milky  Way  in  the  neighborhood  of  the  south  pole. 

597.  Original  condition  of  the  universe. — The  question  naturally 


ORIGINAL   CONDITION   OF  THE   UNIVERSE,  ETC.  313 

arises,  Was  the  universe  created  substantially  as  we  now  see  it, 
or  has  it  been  brought  to  its  present  condition  by  a  succession  of 
gradual  changes  under  the  operation  of  general  laws  ?  We  find 
in  our  solar  system  several  remarkable  coincidences  which  we  can 
not  well  suppose  to  be  fortuitous,  and  which  naturally  suggest  the 
idea  of  some  grand  and  comprehensive  law. 

1st.  All  the  planets  (now  90  in  number)  revolve  about  the  sun 
from  west  to  east,  and,  with  slight  exceptions,  nearly  in  the  same 
plane,  viz.,  the  plane  of  the  sun's  equator.  There  are  only  4  plan- 
ets (and  these  are  minute  asteroids)  whose  orbits  are  inclined  to 
the  ecliptic  as  much  as  20°. 

2d.  The  sun  rotates  on  an  axis  in  the  same  direction  as  that  in 
iffhich  the  planets  revolve  around  him. 

3d.  All  the  major  planets  (except  perhaps  Uranus  and  Nep- 
tune) rotate  on  their  axes  in  the  same  direction  as  that  in  which 
they  move  around  the  sun. 

4th.  The  satellites  (as  far  as  known)  revolve  around  their  pri- 
maries in  the  same  direction  in  which  the  latter  turn  on  their 
axes. 

5th.  The  orbits  of  all  the  larger  planets  and  their  satellites  have 
small  eccentricity.  Only  seven  of  the  asteroids  have  an  eccen- 
tricity as  great  as  one  quarter. 

6th.  The  planets,  upon  the  whole,  increase  in  density  as  they 
are  found  nearer  the  sun. 

7th.  The  orbits  of  the  comets  have  usually  great  eccentricity, 
and  have  every  variety  of  inclination  to  the  ecliptic. 

These  coincidences  are  not  a  consequence  of  the  law  of  univer- 
sal gravitation,  yet  it  is  highly  improbable  that  they  should  be  the 
result  of  chance.  They  seem  rather  to  indicate  the  operation  of 
some  uniform  law.  Can  we  discover  any  law  from  which  these 
coincidences  would  necessarily  result? 

598.  Conclusion*  from  geological  phenomena. — An  examination 
of  the  condition  and  structure  of  the  earth  has  led  geologists  to 
conclude  that  our  entire  globe  was  once  liquid  from  heat,  and  that 
it  has  gradually  cooled  upon  its  surface,  while  a  large  portion  of 
the  interior  still  retains  much  of  its  primitive  heat  The  shape 
of  the  mountains  in  the  moon  seems  to  indicate  that  that  body 
has  at  some  former  time  been  in  a  state  of  fusion.  But  if  the 
earth  and  moon  were  ever  subjected  to  such  a  heat,  it  is  proba- 


314  ASTRONOMY. 

ble  tliat  the  other  members  of  the  solar  system  were  in  a  like 
condition,  perhaps  at  a  temperature  sufficient  to  volatilize  every 
solid  and  liquid  body,  constituting  perhaps  a  single  nebulous  mass 
of  the  smallest  density. 

599.  The  nebular  hypothesis  stated. — Let  us  suppose,  then,  that 
ihe  matter  composing  the  entire  solar  system  once  existed  in  the 
condition  of  a  single  nebulous  mass,  extending  beyond  the  orbit 
of  the  most  remote  planet.     Suppose  that  this  nebula  has  a  slow 
rotation  upon  an  axis,  and  that  by  radiation  it  gradually  cools, 
thereby  contracting  in  its  dimensions.     As  it  contracts  in  its  di- 
mensions, its  velocity  of  rotation,  according  to  the  principles  of 
Mechanics,  must  necessarily  increase,  and  the  centrifugal  force 
thus  generated  in  the  exterior  portion  of  the  nebula  would  at 
length  become  equal  to  the  attraction  of  the  central  mass. 

This  exterior  portion  would  thus  become  detached,  and  revolve 
independently  as  an  immense  zone  or  ring.  As  the  central  mass 
continued  to  cool  and  contract  in  its  dimensions,  other  zones 
would  in  the  same  manner  become  detached,  while  the  central 
mass  continually  decreases  in  size  and  increases  in  density. 

The  zones  thus  successively  detached  would  generally  break 
up  into  separate  masses  revolving  independently  about  the  sun ; 
and  if  their  velocities  were  slightly  unequal,  the  matter  of  each 
zone  would  ultimately  collect  in  a  single  planetary,  but  still  gase- 
ous mass,  having  a  spheroidal  form,  and  also  a  motion  of  rotation 
about  an  axis. 

As  each  of  these  planetary  masses  became  still  farther  cooled, 
it  would  pass  through  a  succession  of  changes  similar  to  those  of 
the  first  solar  nebula ;  rings  of  matter  would  be  formed  surround- 
ing the  planetary  nucleus,  and  these  rings,  if  they  broke  up  into 
separate  masses,  would  ultimately  form  satellites,  revolving  about 
their  primaries. 

600.  Phenomena  explained  ly  this  hypothesis. — The  planet  Saturn 
affords  the  only  instance  in  the  solar  system  in  which  these  rings 
have  preserved  their  unbroken  form ;  and  the  group  of  asteroids 
between  Mars  and  Jupiter  presents  a  case  in  which  a  ring  broke 
up  into  a  large  number  of  small  fragments,  which  continued  to 
revolve  in  independent  orbits  about  the  sun. 

The  first  six  of  the  phenomena  mentioned  in  Art.  597  are  ot> 


THE  NEBULAR  HYPOTHESIS.  315 

vious  consequences  of  this  theory.  The  eccentricity  of  some  of 
the  orbits,  and  their  inclination  to  the  sun's  equator,  must  be  as- 
cribed to  the  accumulated  effect  of  the  disturbing  action  of  the 
planets  upon  each  other. 

601.  Apparent  anomalies  explained. — The  planets  thus  formed 
would  all  have  a  motion  of  rotation,  but  they  would  not  all  nec- 
essarily rotate  in  the  same  direction  as  the  motion  of  revolution. 
The  outer  planets  might  rotate  in  the  contrary  direction,  but  the 
satellites  must  in  all  cases  revolve  in  their  orbits  in  the  same  di- 
rection as  the  rotation  of  the  primary.     The  satellites  of  Uranus 
and  Neptune  have  a  retrograde  motion ;  and  if  it  shall  be  discov- 
ered that  these  planets  rotate  upon  their  axes  in  the  same  direc- 
tion, these  movements  would  all  be  consistent  with  the  nebular 
hypothesis. 

Comets  may  consist  of  nebulous  matter  encountered  by  the 
solar  system  in  its  motion  through  space,  and  thus  brought  within 
the  attractive  influence  of  the  sun.  They  are  thus  compelled  to 
move  in  orbits  around  the  sun,  and  these  orbits  may  become  so 
modified  by  the  attraction  of  the  planets  that  they  may  some- 
times become  permanent  members  of  our  solar  system.  Some 
of  the  comets  may  perhaps  consist  of  small  portions  of  nebulous 
matter  which  became  detached  in  the  breaking  up  of  the  planet- 
ary rings,  and  continued  to  revolve  independently  about  the  sun. 

602.  How  this  hypothesis  may  be  tested. — It  has  been  attempted 
to  subject  this  hypothesis  to  a  rigorous  test  in  the  following  man- 
ner.    The  time  of  revolution  of  each  of  the  planets  ought  to  be 
equal  to  the  time  of  rotation  of  the  solar  mass  at  the  period  when 
its  surface  extended  to  the  given  planet.     It  remains,  then,  to 
compute  what  should  be  the  time  of  rotation  of  the  solar  mass 
when  its  surface  extended  to  each  of  the  planets.     It  has  been 
found  that  if  we  suppose  the  sun's  mass  to  be  expanded  until  its 
surface  extends  to  each  of  the  planets  in  succession,  its  time  of 
rotation  at  each  of  these  instants  would  be  very  nearly  equal  to 
the  actual  time  of  revolution  of  the  corresponding  planet ;  and 
the  time  of  rotation  of  each  primary  planet  corresponds  in  like 
manner  with  the  time  of  revolution  of  its  different  satellites. 

The  nebular  hypothesis  must  therefore  be  regarded  as  possess- 
ing considerable  probability,  since  it  accounts  for  a  large  number 
of  circumstances  which  hitherto  had  remained  unexplained. 


316  ASTRONOMY. 


Miscellaneous  Problems. 

1.  At  what  hour  does  the  sun  rise  at  Havana,  Lat.  23°  9',  at 
the  time  of  the  winter  solstice  ? 

2.  What  is  the  greatest,  and  also  the  least  meridian  altitude  of 
the  sun  at  Chicago,  Lat.  41°  52'  ? 

3.  What  is  the  least  latitude  in  which  twilight  lasts  all  night 
at  the  time  of  the  summer  solstice  ? 

4.  In  what  azimuth  does  the  sun  rise  at  Boston,  Lat.  42°  21',  on 
the  10th  of  May,  when  his  declination  is  17°  45'  N.  ? 

5.  At  what  hour  of  the  day  is  the  sun  due  east  at  New  York, 
Lat.  40°  42',  on  the  10th  of  August,  when  his  declination  is  15° 
26' N.? 

6.  Find  the  duration  of  twilight  at  Cincinnati,  Lat.  39°  6',  on 
the  21st  of  January,  when  the  sun's  declination  is  20°  S. 

7.  Find  the  latitude  of  the  place  where  the  sun's  centre  remains 
above  the  horizon  for  a  hundred  successive  days. 

8.  At  Washington,  Lat.  38°  54',  on  the  1st  of  May,  when  the 
sun's  declination  is  15°  14'  N.,  the  length  of  the  shadow  cast  by 
a  tower  at  noon  on  a  horizontal  plane  is  m  feet;  determine  the 
height  of  the  tower. 

9.  At  New  Haven,  Lat  41°  18',  on  the  20th  of  May,  when  the 
sun's  declination  is  20°  6'  N.,  at  what  hour  of  the  day  will  a  man's 
shadow  be  double  his  height? 

10.  Find  the  altitude  of  the  sun  at  Philadelphia,  Lat.  39°  57', 
on  the  day  of  the  equinox  at  9  o'clock  in  the  morning. 

11.  Find  the  time  of  sunrise  on  the  longest  day  at  a  place  in 
Lat  45°. 

12.  Determine  the  latitude  of  the  place  in  which  the  longest 
day  contains  16  hours. 

13.  Find  the  sun's  altitude  at  6  o'clock  in  terms  of  the  latitude 
of  the  place,  and  declination  of  the  sun. 

14.  Find  the  sun's  altitude  when  on  the  prime  vertical  in  terms 
of  the  latitude  and  declination. 

15.  The  sun's  altitude  at  6  o'clock  was  14°,  and  its  altitude 
when  due  east  was  23° ;  required  the  latitude  of  the  place. 

16.  Determine  the  declination  of  the  sun  that  it  may  set  in  the 
S.W.  point  at  a  place  whose  latitude  is  65°  N. 

17.  Determine  the  latitude  of  the  place  where  the  sun  rises  in 


MISCELLANEOUS  PROBLEMS.  317 

the  KE.  point,  and  also  the  time  of  its  rising,  the  sun's  declina- 
tion being  20°  N. 

18.  The  sun's  meridian  altitude  is  66°,  and  his  depression  below 
the  horizon  at  midnight  is  30°;  required  the  sun's  declination 
and  the  latitude  of  the  place. 

19.  The  longitude  of  Sirius  on  the  1st  of  January,  1864,  was 
101°  1'  10" ;  what  was  its  longitude  at  the  commencement  of  the 
Christian  era,  allowing  50".24  for  mean  amount  of  precession  ? 

20.  In  the  year  1880  there  were  five  Sundays  in  the  month  of 
February;  when  will  a  similar  case  happen  again? 

21.  How  much  faster  than  at  present  must  the  earth  rotate  upon 
its  axis  in  order  that  bodies  on  its  surface  at  the  equator  may  lose 
half  their  weight? 

22.  How  much  faster  than  at  present  must  the  earth  rotate  upon 
its  axis  in  order  that  bodies  on  its  surface,  in  Lat.  60°,  may  lose 
all  their  gravity? 

23.  Determine  the  latitude  of  the  place  where  the  longest  day 
is  6  hours  and  12  minutes  longer  than  the  shortest  day. 

24.  Determine  the  latitude  of  the  place  at  which  the  sun  sets  at 
10  o'clock  on  the  longest  day,  and  also  find  the  latitude  of  the 
place  where  it  sets  at  3  o'clock  on  the  shortest  day. 

25.  Aldebaran  (Dec.  16°  14'  N.)  was  observed  when  on  thb 
prime  vertical  both  east  and  west,  and  the  intervening  time  was 
9h.20m. ;  required  the  latitude  of  the  place. 

26.  Determine  the  latitude  of  the  place  at  which  the  sun  rises  in 
the  N.N.E.  point  at  the  summer  solstice. 

27.  At  a  place  in  Lat  38°,  when  the  sun's  declination  was  20° 
N.,  the  sun  was  observed  to  rise  at  a  point  E.  by  N.  according  to 
a  surveyor's  compass ;  required  the  variation  of  the  needle. 

•  28.  The  horizontal  refraction  being  34'  54",  find  how  much  the 
rising  of  the  sun  is  accelerated  by  it  at  New  Haven  at  the  time 
of  the  summer  solstice. 

29.  Prove  that  the  sun's  rising  is  least  accelerated  by  refraction 
at  the  time  of  the  equinoxes. 

30.  Supposing  the  quantity  of  matter  in  the  sun  to  be  increased 
nine  times,  and  the  orbits  of  the  planets  to  continue  the  same,  how 
would  the  periodic  times  be  altered  ? 

31.  If  the  mass  of  a  planet  be  4  times  that  of  the  earth,  and  the 
distance  of  its  satellite  16  times  that  of  the  moon  from  the  earth, 
in  what  time  will  the  satellite  make  one  revolution? 


318  ASTRONOMY. 

32.  If  the  periodic  time  of  Mercury  be  to  that  of  the  earth  as  4 
to  17,  determine  the  time  of  one  synodic  revolution. 

33.  What  must  be  the  relation  of  the  distances  from  the  sun  of 
a  superior  and  an  inferior  planet  that  their  synodic  revolutions 
may  be  equal  ? 

34.  Determine  when  Saturn  will  appear  stationary,  assuming 
his  distance  from  the  sun  to  be  to  that  of  the  earth  from  the  sun 
as  19  to  2,  and  the  orbits  to  be  circles. 

35.  How  high  must  a  man  be  elevated  above  the  surface  of  the 
earth  at  New  York,  Lat.  40°  42',  to  see  the  sun  at  midnight  at  the 
time  of  the  summer  solstice  ? 

36.  A  place  in  Lat.  42°  has  its  horizon  so  surrounded  with, 
mountains  that  the  sun  is  not  visible  until  it  is  10°  above  the  ra- 
tional horizon  in  the  morning,  and  it  again  disappears  when  10° 
above  the  rational  horizon  in  the  evening;  how  much  is  the  lon- 
gest day  shortened  by  this  circumstance  ? 

37.  At  a  place  in  Lat.  35°  K,  Aldebaran  (R.  A.  67°  2',  Dec.  16° 
14'  N.)  was  seen  in  the  same  vertical  plane  with  Sirius  (R.  A.  99° 
47',  Dec.  16°  32'  S.) ;  required  the  azimuth. 

38.  At  a  place  in  Lat.  35°  K,  find  the  hour  at  which  Aldebaran 
and  Sirius  will  be  in  the  same  azimuth  on  the  1st  of  January, 
when  the  sun's  R.  A.  is  18h.  45m. 

39.  Aldebaran  and  Sirius  were  found  to  set  at  the  same  instant ; 
required  the  latitude  of  the  place  of  observation. 

40.  Find  the  azimuth  of  a  star  when  its  change  of  altitude  in  a 
given  time  is  a  maximum. 

41.  Find  at  what  time  on  the  longest  day  of  the  year, the  vari- 
ation of  the  sun's  altitude  at  New  Haven  is  the  most  rapid. 

42.  Given  the  sun's  apparent  diameter,  and  the  latitude  of  the 
place,  it  is  required  to  determine  his  declination  when  the  time  of 
rising  of  the  sun's  disc  is  a  minimum. 

43.  Find  the  time  when  the  apparent  diurnal  motion  of  a  Ursas 
Majoris  (Dec.  62°  29')  is  perpendicular  to  the  horizon  at  New 
Haven. 

44.  Compare  the  times  during  which  a  Ursa?  Majoris  moves 
eastward  and  westward  at  New  Haven. 

45.  Find  the  sun's  longitude,  or  the  day  of  the  year,  when  Sirius 
rises  with  the  sun  at  a  place  in  Lat.  42°. 

46.  Find  the  day  of  the  year  when  Sirius  sets  with  the  sun  at  a 
place  in  Lat.  42°. 


MISCELLANEOUS  PROBLEMS. 


319 


47.  When  the  sun's  declination  was  15°  N,  his  altitude  was 
found  to  be  20°,  and  after  one  hour's  interval  his  altitude  was 
found  to  be  31° ;  required  the  latitude  of  the  place  of  observation. 

48.  If  the  length  of  the  day  be  to  that  of  the  night  as  3  to  2, 
and  the  altitude  of  the  sun  at  noon  double  his  depression  at  mid- 
night, determine  the  latitude  of  the  place,  and  the  sun's  declina- 
tion. 

49.  Determine  at  what  place  and  at  what  time  of  the  year,  day 
breaks  at  2  o'clock,  and  the  sun  rises  at  half  past  four. 

50.  Find  the  sun's  declination  when  the  twilight  is  shortest  at 
New  York. 

51.  If  the  moon  has  a  small  satellite  whose  periodic  time  is  five 
days,  what  must  be  the  major  axis  of  its  orbit? 

52.  It  is  required  to  find  the  sidereal  period  of  Mercury,  the 
position  of  the  ascending  node,  and  the  distance  of  Mercury  from 
the  sun,  from  the  following  data  : 


Green'h  Time. 
183& 

Places  of 
Geocentric  A.  R. 

Mercury. 
Geocentric  Dec. 

Sun's  longitude. 

Earth  H  Radius 
Vector. 

Feb.  1  5 
"    16 
May  1  3 
"    i4 

h.      in.       s. 

22  55  19.8 

23      0    Sg.O 

3  i3  4i.5 

3    22    20.9 

6  59  55  S. 
6  i3     4  S. 
17  53  56  N. 

18  3y  5i  N. 

326      8      4 

327     8  37 
52     57  10 
53     55     o 

o  .  988071 

0.988284 

i  .oino4 

i  .oii323 

Obliquity  of  the  ecliptic  a3°  27'  16" 


THE  following  Alphabet  is  given  in  order  to  facilitate,  to  the  student  who  is  un- 
acquainted with  it.  the  reading  of  those  parts  in  which  the  Greek  letters  are  used  : 


Letter*. 

Name,, 

Letters. 

Names. 

A 

a 

Alpha. 

N 

V 

Nu. 

B 

ft 

BSta 

2 

f 

Xi. 

r 

7 

Gamma. 

0 

o 

Omicron. 

A 

9 

Delta. 

n 

C7  TT 

Pi. 

E 

e 

Epsilon. 

p 

P 

Rho. 

Z 

ff 

Zeta. 

2 

CT  f 

Sigma. 

H 

T\ 

Eta. 

T 

r 

Tau. 

0 

40 

ThEta. 

T 

11 

Upsilon. 

I 

i 

Iota. 

«J> 

* 

Phi. 

K 

K 

Kappa. 

X 

* 

Chi. 

A 

A 

Lambda. 

* 

f 

Psi. 

M 

^ 

KB. 

Q 

H 

Omega. 

TABLE   I.— ELEMENTS   OF  THE   PLANETARY  SYSTEM.       321 


Name. 

=  _j                       Distance 

frc 

m  the  Sun. 

liccentricitv. 

Sidereal 
Revolution 

a-1            Mean. 

Greatest. 

Least. 

in  Day?.     • 

Mercury.  . 

$           0.38710 

O 

.46669 

0.30750 

0.20662 

87. 

969 

Venus  .  .  . 

9       0.72333 

O 

.72826 

0.7l84o 

.oo683 

224- 

701 

Earth  

6        i  .  ooooo 

I 

.01678 

0.98322 

.01677 

365. 

266 

Mars  

$        1.62369 

I 

.66578 

i.38i6o 

.09326 

686. 

980 

Jupiter  .  .  . 

H       6.20280 

5 

.45378 

4.96182 

.04824 

4332. 

585 

Saturn  .  .  . 

»?       g.53885 

10 

.07328 

9.00442 

.06600 

10769. 

220 

Uranus  .  .  . 

$     19.18264 

20 

.07612 

18.28916 

.04668 

3o686. 

821 

Neptune.  . 

*      3o.o3697 

3o 

.2 

9888 

29.77606 

.00872 

60126. 

722 

Name. 

Synodic*! 
Revolution 
in  Days. 

Mean  daily 
Motion. 

Hourly 
Motion 
in  Miles. 

Inrlinntion 
of  Orbit. 

Light  at 

pres- 
sion. 

Perihelion. 

Aphelion. 

Mercury  .  . 

Il5.877 

245 

32.6 

109796 

708 

IO 

.576 

4.692 

-Hbv 

Venus  .  .  . 

583.  921 

96 

7- 

s 

8o320 

3  23  3i 

I 

1.886 

Earth  

59 

8.3 

683n 

I 

.o34 

0.967 

g|^ 

Mars  .... 

779.936 

3i 

26. 

7 

5534i 

i   5i     5 

O 

.624 

o.36o 

& 

Jupiter.  . 

398.884 

4 

59. 

3 

29948 

i   1  8  4o 

0 

.o4i 

o.o34 

Saturn  .  .  . 

378.092 

2 

o. 

6 

22118 

2    29    28 

0 

.012 

O.OIO 

A 

Uranus  .  .  . 

369.656 

42. 

4 

16697 

o  4 

6  3o 

o 

.oo3 

O.002 

Neptune  .  . 

367.48q 

21  . 

6 

12464 

i  46  69 

o 

.001 

O.  OO 

Name. 

Time  of 
Rotation. 

Equ 

atorial  Diam- 
eter. 

Volume. 

Mass. 

Dens- 
ity. 

Grav- 
ity. 

Bodies 
fall  In  one 
Second. 

Apparent. 

In 

Miles. 

Sun   
Mercury  .  . 

h.     m.    B. 
600 

24    5  28 

1923.6 

6.7 

888000 

3ooo 

i4i6ooo 
0.069 

354936 
0.118 

0.26 
2.01 

28.56 

0.53 

Feet. 

459.2 

8.5 

Venus  .  .  . 

23    21     21 

16.6 

7700 

0.912 

0.883 

0.97 

0.92 

i4.8 

Earth  

23  56     4 

7926 

i 

.000 

I  .000 

I  .00 

i  .00 

16.1 

Mars  
Jupiter.  .  . 
Saturn  .  .  . 

24    37    22 

9  55  26 
10  29   17 

38.3 
17.0 

45oo 
92000 
76000 

o.i83 

l4l2.0 
770.0 

O.I  32 

338.034 
101  .o64 

0.72 

0.24 

o.i3 

0.62 
2.70 
1.19 

8.4 

43.4 
19.1 

Uranus  .  .  . 

i 

',    i 

3 

6000 

96.9 

14.789 

0.  ID 

0.76 

2.1 

Neptune.  . 

2.6 

35ooo 

89.6 

24.648 

0.27 

0.85 

i3.7 

322 


TABLE  II. — THE  MINOR  PLANETS. 


No. 

Name. 

Date  of  Discovery. 

Discoverer. 

Mean 
Distance. 

Sidereal 
Revolution 
in  Days. 

Eccen- 
tricity. 

j 

Ceres 

1801,  Jan.  i 

Piazzi 

2.7667 

1680.9 

0.0802 

2 

Pallas 

1802,  March  28 

Olbers 

2.7696 

1683.5 

.24oO 

3 

Juno 

1804,  Sept.  i 

Harding 

2.6680 

1691.7 

.2573 

4 

Vesta 

1807,  March  29 

Olbers 

2.36i3 

i325.3 

.0898 

5 

Astroea 

1  845,  Dec.  8 

Hencke 

2.5765 

i5io.6 

.1900 

6 

Hebe 

1847.  July  i 

Hencke 

2.425l 

1379.4 

.2030 

7 

Iris 

1  647,  Aug.  1  3 

Hind 

2.3863 

1346.5 

.23i3 

8 

Flora 

1  84?,  Oct.  18 

Hind 

2.20l4 

1192.9 

.1667 

9 

Metis 

1  848,  April  25 

Graham 

2.3858 

i346.o 

.  1  24  1 

10 

Hygeia 

1849,  April  12 

Gasparis 

3.i5n 

2o43.  i 

.1000 

ii 

Parthenope 

i85o,  May  11 

Luther 

2.4525 

1402.9 

.0993 

12 

Victoria 

i85o,  Sept.  1  3 

Hind 

2.3342 

i3o2.6 

.2189 

i3 

Egeria 

i85o,  Nov.  2 

Gasparis 

2.5763 

1610.4 

.0867 

i4 

Irene 

1  85  1,  May  19 

Hind 

2.5895 

1622.0 

.1662 

i5 

Eunomia 

1  85  1,  July  29 

Gasparis 

2.6437 

1670.0 

.1872 

16 

Psyche 

1  852,  March  17 

Gasparis 

2.9264 

1828.6 

.i34i 

i«7 

Thetis 

1  852,  April  17 

Luther 

2.4733 

1420.7 

.1276 

18 

Melpomene 

1862,  June  24 

Hind     * 

2.2966 

1270.4 

.2177 

19 

Fortuna 

I  852,  Aug.  22 

Hind 

2.44i6 

i393.5 

.1672 

20 

Massilia 

1  852,  Sept.  19 

Gasparis 

2.4089 

1365.6 

.i443 

21 

Lutetia 

1  852,  Nov.  1  5 

Goldschmidt 

2.4354 

1388.2 

.1620 

22 

Calliope 

1  8  52,  Nov.  1  6 

Hind 

2.9092 

1812.4 

.1010 

23 

Thalia 

i852,Dec.  1  5 

Hind 

2-6271 

i555.3 

.2323 

24 

Themis 

1  853,  AprilS 

Gasparis 

3.1420 

2034.2 

.1170 

25 

Phocaea 

a  8  5  3,  April  7 

Chacornac 

2.4008 

1358.8 

.2546 

26 

Proserpina 

1  853,  May  5 

Luther 

2.656i 

i58i.i 

.0876 

27 

Euterpe 

1  853.  Nov.  8 

Hind 

2.3468 

i3i3.2 

.1735 

28 

Bellona 

1  8  54  \  March  i 

Luther 

2.7785 

1691  .6 

.  i5oi 

29 

Amphitrite 

1  854,  March  i 

Marth 

2.5544 

i4gi  .2 

.o739 

3o 

Urania 

1  854,  July  22 

Hind 

2.3663 

i329.6 

.1270 

3i 

Euphrosyne 

1  854,  Sept.  i 

Ferguson 

3.1571 

2049.0 

.2181 

32 

Pomona 

1  854,  Oct.  26 

Goldschmidt 

2.5868 

1619.6 

.0824 

33 

Polyhymnia 

1  854,  Oct.  28 

Chacornac 

2.865: 

1771.3 

.3382 

34 

Circe 

i855,April6 

Chacornac 

2.6871 

1608.9 

.1066 

35 

Leucothea 

1  855,  April  19 

Luther 

S.ooSg 

I9o3.5 

,ai4i 

36 

Atalanta 

1  855,  Oct.  5 

Goldschmidt 

2.7450 

1661.  i 

.3007 

3? 

Fides 

1  855,  Oct.  5 

Luther 

2.64i3 

1668.0 

.1767 

38 

Leda 

1  8  56,  Jan.  12 

Chacornac 

2.7401 

1656.8 

.i554 

39 

Lajtitia 

1  856,  Feb.  8 

Chacornac 

2.7700 

i683.9 

.11  ii 

4o 

Harmon  ia 

1  856,  March  3i 

Goldschmidt 

2.2677 

1247.3 

.o463 

4i 

Daphne 

1  856,  May  22 

Goldschmidt 

2.7674 

i68i.5 

.2703 

42 

Isis 

i856,  May  23 

Pogson 

2.4400 

1392.2 

.2256 

43 

Ariadne 

1857.  April  1  5 

Pogson 

2.2034 

ng4.6 

.1676 

44 

Nysa 

i857;May27 

Goldschmidt 

2.4234 

1378.0 

.i5o3 

45 

Eugenia 

1857.  June  27 

Goldschmidt 

2.7205 

1639.0 

.0822 

TABLE  II. — THE  MINOR  PLANETS. 


323 


No. 

Name. 

Date  of  Discovery. 

Discoverer. 

Mean 
Distance. 

Siuereal 
Revolution 
in  I>»VH. 

Kccen- 
tricity. 

46 

47 

Hestia 
Aglaia 

1857,  Aug.  1  6 
1857,  Sept.  1  5 

Popson 
Luther 

2.526l 
2.8802 

1466.5 
i785.4 

O.l647 

.i324 

48 

i9 

5o 

Doris 
Pales 
Virginia 

1857,  Sept.  19 
i857,Sept.  19 
i857,0ct.4 

Goldschmidt 
Goldschraidt 
Ferguson 

3.io95 
3.o84o 
2.6497 

2002.8 

19-78.2 
i575.4 

.0766 
.2374 
.2873 

5i 

Nemausa 

1  858,  Jan.  22 

Laurent 

2.3655 

1328.9 

.0667 

52 

Europa 

1  858,  Feb.  6 

Goldschmidt 

3.1000 

i  oo3  .  6 

«uvv  i 

•  i  '  1  1  4. 

53 

Calypso 

1  858,  April  4 

Luther 

2.6188 

1  547  .0 

.2087 

54 

Alexandra 

i858,  Sept.  10 

Goldschmidt 

2.7123 

^  /    V 

i63i.6 

•  *u«j  i 

.  1969 

55 

Pandora 

1  858',  Sept.  10 

Searle 

2.7590 

i673.9 

.1447 

56 

Melete 

i857,  Sept.  9 

Goldschmidt 

2.5971 

i528.7 

.2370 

57 

Mnemosyne 

1859,  Sept.  22 

Luther 

3.i573 

2049.1 

58 

Concordia 

1860.  March  24 

Luther 

2.6950 

1616.  o 

.o4o3 

59 

Elpis 

1860^  Sept.  12 

Chacornac 

2.7132 

i632.4 

.1171 

60 

Echo 

1860,  Sept.  1  5 

Ferguson 

2.393i 

1352.2 

.i847 

61 

Danae 

1860,  Sept.  19 

Goldschmidt 

2.9854 

i884.i 

,i65i 

62 

Erato 

1860,  Oct. 

Forster 

3.1287 

2021  .4 

.1762 

63 

Ausonia 

1861,  Feb.  10 

Gasparis 

2.3957 

i354.4 

.1254 

64 

Angelina 

1  86  1,  March  4 

Tempel 

2.6807 

i6o3.2 

.  1295 

65 

Cybele 

1  86  1,  March  8 

Tempel 

3.4209 

23II.O 

.  1204 

66 

Maia 

i86i.April9 

Tuttle 

2.6635 

i587.8 

.1339 

67 

Asia 

1861,  April  17 

Pogson 

2.4209 

i375.8 

.1844 

68 

Leto 

1861,  April  29 

Luther 

2.7822 

i695.6 

.i875 

69 

Hesperia 

1861.  April  29 

Schiaparelli 

2.9950 

1893.1 

.1745 

70 

Panopasa 

i86ijMay  5 

Goldschmidt 

2.6129 

i  542.  7 

.i83o 

7i 

Feronia 

1  86  1.  May  29 

Peters 

2.2654 

1245.4 

.1195 

72 

Niobe 

1  86  1,  Aug.  1  3 

Luther 

2.7562 

1671.3 

.1737 

73 

Clytie 

1862,  April  7 

Tuttle 

2.6655 

i589.6 

.o454 

74 

Galatea 

1862.  Aug.  29 

Tempel 

2.7785 

1691.7 

.2384 

75 

Eurydice 

1862^  Sept.  22 

Peters 

2.6708 

i594.3 

.3o67 

76 

Freia 

1862,  Oct.  21 

D'Arrest 

3.1890 

2080.0 

.0302 

77 

Frigga 

1862.  Nov.  12 

Peters 

2.6738 

1596.9 

.1358 

78 

Diana 

1  863',  March  i5 

Luther 

2.6263 

i554.6 

.2067 

79 

Eurynome 

1  863.  Sept.  i4 

Watson 

2.4478 

i398.8 

.I932 

80 

Sappho 

1864',  May  2 

Pogson 

2.2971 

1271  .6 

.2005 

81 

Terpsichore 

1  864,  Sept.  3o 

Tempel 

2.7800 

i693.o 

.i3i3 

82 

Alcmene 

1  864,  Nov.  27 

Luther 

2.7427 

i659.i 

.1981 

83 

Beatrix 

i865^  April  26 

Gasparis 

2.4288 

1382.5 

.0842 

84 

Clio 

i865,  Aug.  25 

Luther 

2  .  36oo 

1324.2 

.2346 

85 

lo 

1  865,  Sept.  19 

Peters 

2.6536 

i578.9 

.1907 

86 

Semele 

1866,  Jan.  4 

Tietjen 

3.0908 

1984.8 

.2049 

87 

Sylvia 

1866,  May  1  6 

Pogson 

3.4927 

2384.1 

.O8ll 

88 

Thisbe 

1866,  June  i5 

Peters 

2.6733 

i596.5 

.1888 

89 

Julia. 

1866,  Aug.  6 

Stephan 

2.5475 

i485.i 

.1672 

9° 

Antiope 

1866,  Oct.  i 

Luther 

3.ii88 

2011  .8 

.1477 

324 


TABLE  II.— THE  MINOR  PLANETS. 


No. 

Name. 

Date  of  Discovery. 

Discoverer. 

Mean 
Distance. 

Sidereal 
Kevolutiou 
in  Days. 

Eccen- 
tricity. 

91 

JEgina 

1866,  Nov.  4 

Stephan 

2.5895 

l522.O 

o.io87 

92 

Undina 

1867,  July  7 

Peters 

3.i85i 

2076.4 

.1024 

93 

Minerva 

1867,  Aug.  24 

Watson 

2.7537 

1669.2 

.i4o5 

94 

Aurora 

1867,  Sept.  6 

Watson 

3.i6o8 

2052-7 

.o87o 

95 

Arethusa 

1867,  Nov.  23 

Luther 

3.o785 

I973.i 

.1455 

96 

JEgle 

1868,  Feb.  17 

Coggia 

3.o497 

i945.  3 

.i4o5 

97 

Clotho 

1868,  Feb.  17 

Tempel 

2.6672 

iSgi.o 

.258i 

98 

lanthe 

1868,  April  1  8 

Peters 

2.6878 

1609.7 

.1896 

99 

Dike 

1868,  May  29 

Borelly 

2.7967 

1708.3 

.2384 

IOO 

Hecate 

1868,  July  n 

Watson 

3.0978 

1991.7 

.i565 

101 

Helena 

1868,  Aug.  1  5 

Watson 

2.5843 

i5i7.6 

.1392 

102 

Miriam 

1868,  Aug.  22 

Peters 

2.6609 

1585.6 

.255i 

io3 

Hera 

1868,  Sept.  7 

Watson 

2.7020 

1622.4 

.o836 

io4 

Clymene 

1868,  Sept.  1  3 

Watson 

3.i5i2 

2043.2 

.i773 

io5 

Artemis 

1868,  Sept.  1  6 

Watson 

2.3761 

i337.9 

.1728 

106 

Dione 

1868,  Oct.  10 

Watson 

3.i6o5 

2052.3 

.1829 

107 

Camilla 

1868,  Nov.  17 

Pogson 

3.56o2 

2453.8 

.1227 

108 

Hecuba 

1869,  April  2 

Luther 

3.2ii3 

2I02.O 

,ioo5 

109 

Felicitas 

1869,001.9 

Peters 

2.6952 

1616.2 

.2988 

no 

Lydia 

1870,  April  19 

Borelly 

2.7402 

i657.5 

.0798 

III 

Ate 

1870,  Aug.  i4 

Peters 

2.5938 

i526.o 

.1060 

112 

Iphigenia 

1870,  Sept.  19 

Peters 

2.4335 

1386.5 

.1286 

ii3 

Amalthea 

1871,  March  1  2 

Luther 

2.3765 

1338.3 

.0878 

n4 

Cassandra 

1871,  July  23 

Peters 

2.6758 

i598.7 

.i4oi 

ii5 

Thyra 

1871,  Aug.  6 

Watson 

2.3791 

i34o-.4 

.I939 

116 

Sirona 

1871,  Sept.  8 

Peters 

2.7661 

i68o.5 

.i439 

117 

Lomia 

1871,  Sept.  12 

Borelly 

2.9907 

1889.1 

.0229 

118 

Peitho 

1  872,  March  i5 

Luther 

2.4377 

i39o.i 

.1646 

119 

Althaea 

1872,  April  3 

Watson 

2.58oo 

i5i3.6 

.0839 

120 

Lachesis 

1872,  April  10 

Borelly 

3.i3i9 

2024.6 

.o544 

121 

Hermione 

1872,  May  12 

Watson 

3.4625 

2353.3 

.1237 

122 

Gerda 

1872,  July  3  1 

Peters 

3.2196 

2110,3 

.0572 

123 

Brunhilda 

1872,  July  3  1 

Peters 

2.6925 

1613.7 

.n33 

124 

Alceste 

1872,  Aug.  23 

Peters 

2.6296 

i557.4 

.0784 

125 

Liberatrix" 

i872j  Sept.  n 

Pr.  Henry 

3.o352 

i93i.5 

.3467 

126 

Velleda 

1  872,  Nov.  5 

Pa.  Henry 

2.4399 

1392.1 

.1061 

:     I27 

Johanna 

1  8  72,  Nov.  5 

Pr.  Henry 

2.7540 

1669.2 

.0627 

128 

Nemesis 

1872,  Nov.  25 

Watson 

2.7500 

i665.9 

.1256 

129 

Antigone 

i873,Feb.  5 

Peters 

2.8757 

i78i.i 

.2076 

i3o 

Electra 

1  873,  Feb.  17 

Peters 

3.1298 

2022.4 

.2O40 

i3i 

Vala 

1  873,  May  24 

Peters 

2.4202 

i375.2 

.0814 

I  32 

JEthra. 

1873,  June  i3 

Watson 

2.6025 

i533.5 

.3820 

;  i33 

Cyrene 

1873,  Aug.  1  6    ' 

Watson 

3.o648 

1959.6 

.1365 

1  34 

Sophrosyne 

i873,  Sept.  27 

Luther 

2.5673 

i5o2.6 

.1172 

.135 

Hertha 

1874.  Feb.  1  8 

Peters 

2.4292 

i383.o 

.  2046 

TABLE  II. — THE  MINOR  PLANETS. 


325 


No. 

Name. 

Date  of  Discovery. 

Discoverer. 

Mean 

Distance. 

Sidereal 
Revolution 
In  Day*. 

Eccen- 
tricity. 

1  36 

Austria 

i874,  March  18 

Palisa 

2.3o35 

1276.9 

3.II32 

i37 

Melibcea 

1874,  April  21 

Palisa 

3.i334 

2025.9      -2o85 

i38 

Tolosa 

1874,  May  19 

Perrotin 

2.4022 

1360.2 

.i34o 

i39 

Jnewa 

1874.  Oct.  10 

Watson 

2.8739 

1779.5 

.o5i3 

i4o 

Siwa 

i874,  Oct.  1  3 

Palisa 

2.7069 

1626.8    | 

.1979 

i4i 

Lumen 

1875.  Jan.  i3 

Pa.  Henrv 

2.7095 

1629.0 

.2233 

i4a 

Polana 

1875!  Jan.  28 

Palisa 

2.3872 

1347-0 

.io54 

i43 

Adria 

1  875;  Feb.  23 

Palisa 

2.7525 

I668.I 

.o635 

1  44 

Vibilia 

1875.  June  3 

Peters 

2.6423 

i568.9 

.2286 

i45 

Adeona 

1875^  June  3 

Peters 

2.6939 

i6i5.o 

.2127 

i46 

Lucina 

1875.  June  8 

Borelly 

2.7077 

1627.6 

.0692 

Protogeneia 

1875!  July  10        Si-hnlhof 

3.1254 

2018.0 

.0296 

1  48 

Gnllia 

1875^  Aug.  7       j  Pr.  Henry 

2.7830 

i695.9 

.1844 

i4g 

Medusa 

1875,  Sept.  21 

Perrotin 

2.1327 

n37.6 

.n94 

ill 

Nuwa 

1875,  Oct.  18 

Watson 

2.9807 

1879.6 

.i376 

i5i 

Abundantia 

1875.  Nov.  i 

Palisa 

2.584i 

i5i7.3 

.o998 

l52 

Atala 

1  875^  Nov.  2 

Pa.  Henry 

3.i320 

2024.6 

.0822 

i53 

Hilda 

1875.  Nov.  2 

Palisa 

3.95o4 

2867.9 

.i63i 

1  54 

Bertha 

1875;  Nov.  6 

Pr.  Henry 

3.2210 

2III  .5 

.  1001 

i55 

Scyllu 

i875,Nov.  8 

Palisa 

i56 

Xantippe 

l875,  NOV.  22 

Palisa 

3.o375 

1933.7 

.2637 

i58 

Dejanira 
Coronis 

1875,  Dec.  i 
1876.  Jan.  4 

Borelly 
Knorre 

2.5857 
2.8687 

i5i8.6 
1774.7 

.2I99 

.o56o 

i5o 

Emilia 

1876,  Jan.  26 

Pa.  Henry 

3.1252 

2018.0 

.ii55 

139 
160 

Una 

1876,  Feb.  20 

Peters 

2.754a 

1669.5 

.o637 

161 
162 
i63 
1  64 

i65 

Athor 
Laurentia 
Erigone 
Eva 
Loreley 

1876,  April  1  6 
1876,  April  21 
1876,  April  26 
1876,  July  12 
1876,  Aug.  9 

Watson 
Pr.  Henry 
Perrotin 
Pa.  Henrv 
Peters 

2.3807 
3.0270 
2.4066 
2.6117 
3.1286 

i34i-7 
1923.5 
1363.8 
i54i.6 

2021.3 

.:38i 
.i737 
.o895 
.3807 
.0726 

1  66 

167 
1  68 
169 
170 

Rhodope 
Urda 
Sibvlla 
Zelia 
Maria 

1876,  Aug.  i5 
1876.  Aug.  28 
1876!  Sept.  27 
1876^  Sept.  28 
1877,  Jan.  10 

Peters 
Peters 
Watson 
Pr.  Henry 
Perrotin 

2.7199 
3.2186 
3.3781 
2.358o 
2.55io 

1638.5 
2109.1 
2267.8 

1322.5 

i488.2 

.2386 
.3119 
.0672 
.i3o6 
.o65i 

171 
172 
I73 
1  74 

i75 

Ophelia 
Baucis 
Ino 
Phredra 
Andromache 

i877,  Jan.  1  3 
1877,  Feb.  5 
1877,  March  2 
1877,  Sept.  2 
1877,  Oct.  i 

Borelly 
Borelly 
Borelly 
Watson 
Watson 

3.1355 
2.3794 
2.7449 
2.8634 
3.5oo4 

2027.9 
i34o.6 
1661.1 
1769.8 
2392.1 

.1264 
.2046 

.i5o9 
.3492 

176 

177 
178 
179 
1  80 

Idnnna 
Irma 
Belisana 
Clytemnestr 
Garumna 
.  —  

1877.  Oct.  i4 
1877^  Nov.  5 
1877.  Nov.  6 
1877,  Nov.  12 
1878.  Jan.  29 


Peters 
Pa.  Henry 
Palisa 
Watson 
Perrotin 

3.1906 

2.7889 
2.4583 
2.9758 
2.7290 

.  ' 

2081.6 
1701.  i 

i4o7.8 
1875.0 
1646.7 

.i64i 
.2469 
.1266 

.I072 
•177° 

326 


TABLE   III.— ELEMENTS   OF  THE   SATELLITES. 


Elements  of  the  Moon. 


Mean  distance  from  the  earth 288900  miles. 

Mean  sidereal  revolution z'jd.  7h.  43m.  1 1 .46s. 

Mean  synodical  revolution 2gd.  I2h.  44m.  2.873. 

Mean  revolution  of  nodes i8yrs.  2i8d.  2ih.  22in.  46s. 

Mean  revolution  of  perigee 8yrs.  3iod.   i3h.  48m.  53s. 

Mean  inclination  of  orbit 5°  8'  48". 

Eccentricity  of  the  orbit o. 064908. 

Diameter  of  the  Moon 2 160  miles. 

Density  of  the  Moon,  that  of  the  earth  being  i .  .0.5667. 
Mass  of  the  Moon,  that  of  the  earth  being  i .  . . .  fa. 


Elements  of  the  Satellites  of  Jupiter. 


Sidereal 
Revolution. 


Distance 

in  Radii  of 

Jupiter. 


Distance 
in  Miles. 


Orbit  inclined 

to  Jupiter's 

KqiiHtor. 


Diameter. 


Apparent.   In  Miles 


MM,  tip* 

of  Jupiter 
being  1. 


d.  h.  m.  a. 

I  18  27  33 

3  i 3  i4  36 

7  3  42  33 

16  16  3i  o 


6.o4g 

9.623 

i5.35o 

26.998 


278642 
442904 
706714 

I20OOOO 


O07 

o  i  6 
o  5  3 
o  o  24 


2436 
2187 
3573 
3o57 


.0000173 

.0000232 
.0000885 
.0000427 


Elements  of  the  Satellites  of  Saturn. 


Date  of 
Discov- 
ery. 


Sidereal 
Revolution. 


in  Radii  of 


Eccentricity 
of  Orbit 


1789 
1787 

1 684 
1 684 
1672 
1666 
1 848 
1671 


il.   h.  m.  s. 

0  22  37  5 

1  8  53  7 

1  21  l8  26 

2  17  4i  9 
4  12  26  it 

16  22  4i  26 

22  12 

79  7  53  4o 


3.i4i 
4.o32 

4-993 

6.399 

8.932 

20.706 

27.786 
64.35g 


26.8 
34-4 
42.6 
54.5 
76.2 
176.6 

222.1 

5i4.5 


118000 
162000 
188000 
240000 
336ooo 
778000 
940000 
2268000 


0.0089 
0.006 i 

.02 

.0227 
.0292 
.116 

.025 


5oo 
5oo 

1200 
2850 

1800 


Elements  of  the  Satellites  of  Uranus. 


Date  of 
Discovery. 


Sidereal 
Revolution. 


I      Distance  in        Mean  uppartnt    Meau  Distance 
Ttadii  of  I'ranus.         Distance.  in  Miles. 


1 847 
1847 
1787 
1787 


d.  h.  m. 

2  12  17 

4  3  28 

8  16  56 

i3  ii  7 


6. 94 

9.72 

16.89 

21.27 


i3.5 
i9.3 
33.9 
45.2 


119994 
170863 
288600 
38oooo 


Elements  of  the  Satellite  of  Neptune. 


Sidereal  revolution 5d.  2oh.  5om.  45s. 

Apparent  mean  distance 1 6" .  98. 

Mean  distance  in  miles 236ooo. 

Orbit  inclined  to  the  plane  of  the  ecliptic 1 5 1  °. 


TABLE  IV.— ECLIPSES  OF  THE  SUN  FROM  1865  TO  1900.   327 


TABLE  IV.  —  ECLIPSES  OF  THE  SUN  FROM  1865  TO  1900. 

The  following  is  a  list  of  all  the  solar  eclipses  visible  in  the  city  of  Boston  from 
1865  to  the  close  of  the  present  century.  The  dates  are  given  in  mean  time  for  the 
meridian  of  Boston,  reckoned  astronomically. 

1.  1865,  October  I8tk  and  19  'h. 

Beginning  ..........................................  18th,  21h.    9m.  55s. 

Greatest  obscuration  .............................  22     44      58 

Apparent  conjunction  ...........................  22     46        4 

End  ..................................................  19th,    0     25         7 

Magnitude  of  the  eclipse  (sun's  diameter  —  1)  0.692,  on  sun's  sonth  limb. 

This  eclipse  was  annular  in  the  States  of  North  and  South  Carolina  ;  at  Charles- 
ton the  ring  lasted  6f  minutes. 

This  was  the  third  return  of  the  eclipse  of  September,  1811,  which  was  antrimr  in 
Virginia. 

2.  1866,  October  7th  and  Sth. 

Beginning  ...........................................  7th,  23h.  lira.  33s. 

Apparent  conjunction  .............................          23     33      50 

Greatest  obscuration  ...............................  23     41       25 

End  ...................................................  8th,    0      10      34 

Magnitude  of  the  eclipse  0.043,  on  sun's  north  limb. 

South  of  Connecticut  there  was  no  eclipse,  and  no  central  eclipse  in  any  part  of 
the  earth. 

3.  1869,  August  7th. 

Beginning  ...................................................  5h.  21m.  17s. 

Apparent  conjunction  ....................................  6      16        7 

Greatest  obscuration  .....................................  6      16      40 

End  ...........................................................  7        7      28 

Magnitude  of  the  eclipse  0.853,  on  sun's  south  limb. 
This  eclipse  was  total  in  Iowa,  Illinois,  Kentucky,  and  North  Carolina. 

4.  1873,  May  25th. 

The  sun  and  moon  will  bo  in  contact  at  sunrise,  but  the  sun  will  be  eclipsed  to 
places  at  a  greater  distance  from  the  equator,  and  in  less  longitude  from  Greenwich. 

5.  1875,  September  28th. 

Sun  rises  eclipsed  ........................................  17h.  56m.    Os. 

Formation  of  the  ring  ...................................  18     20      21 

Apparent  conjunction  ...................................  18 

Nearest  approach  of  centres  ...........................  18     21      37 

Rupture  of  the  ring  ......................................  18     22      52 

End  of  the  eclipse  ........................................  19     30      43 

Magnitude  of  the  eclipse  at  sunrise,  0.603  ;  at  nearest  approach,  0.951. 
This  eclipse  will  be  annular  in  Boston,  and  in  some  part  of  Maine,  New  Ham]> 
shire,  Massachusetts,  and  Vermont. 

C  1876,  March  25th. 

Beginning  ........  .  ..........................................  *h.  llm.  29s. 

Greatest  obscuration  ......................................  >       * 

•> 


Apparent  conjunction 


End 


48 


Magnitude  of  the  eclipse  0.276,  on  sun's  north  limb. 


328     TABLE  IV. — ECLIPSES  OF  THE  SUN  FROM  1865  TO  1900. 

7.  1878,  July  29th. 

Beginning 4h.  56m.  10s. 

Greatest  obscuration 5      50        1 

Apparent  conjunction 5      53      57 

End 6      39        8 

Magnitude  of  the  eclipse  0.615,  on  sun's  south  limb. 

This  eclipse  will  be  total  at  Denver,  Colorado,  and  the  line  of  central  eclipse  runs 
from  northwest  to  southeast  across  the  western  portion  of  the  United  States. 
This  is  the  fourth  return  of  the  total  eclipse  of  June  IGth.  1806. 

J.  1880,  December  30th. 

Sun  rises  eclipsed 19h.  30m.    Os. 

Greatest  obscuration 20     12      50 

Apparent  conjunction 20     12      59 

End 21      11       37 

Magnitude  of  the  eclipse  at  sunrise,  0230;   at  greatest  obscuration,  0.457,  on 
sun's  north  limb.     This  eclipse  can  not  be  central  in  any  place. 

At  the  time  of  this  eclipse  the  sun  and  moon  are  very  nearly  at  their  least  possi* 
ble  distance  from  the  earth. 

9.  1885,  March  16th. 

Beginning Oh.  35m.    Os. 

Greatest  obscuration 1     55      55 

Apparent  conjunction 1     57      22 

End 3     10      49 

Magnitude  of  the  eclipse  0.537,  on  sun's  north  limb. 

10.  1886,  Avgust  28th. 

Beginning.. 18h.  30m.  22s. 

Apparent  conjunction 18     33       15 

Greatest  obscuration 18     40        1 

End 18     5L      52 

Magnitude  of  the  eclipse  0.018,  on  sun's  south  limb. 

North  of  Massachusetts  there  will  be  no  eclipse. 

11.  1892,  October  20th. 

Beginning Oh.  18m.  39s. 

Apparent  conjunction 1     45       11 

Greatest  obscuration 1     51        8 

End 3     20        8 

Magnitude  of  the  eclipse  O.G82,  on  sun's  north  limb. 

The  sun  will  probably  be  centrally  eclipsed  in  the  Canadas  and  Labrador. 

12.  1897,  July  28th. 

Beginning 21h.    7m.  35s. 

Greatest  obscuration 22      15      35 

Apparent  conjunction 22     24      56 

End 23     23      59 

Magnitude  of  the  eclipse  0.369,  on  sun's  south  limb. 

13.  1900,  May  27th. 

Beginning 20h.    8m.  41s.         > 

Apparent  conjunction 21     22      50 

Greatest  obscuration 21     23        6 

End 22     45      32 

_     Magnitude  of  the  eclipse  0.918,  on  sun's  south  limb. 

The  sun  will  be  totally  eclipsed  in  the  State  of  Virginia. 


TABLE  V. — CYCLE  OF  ECLIPSES. 


329 


Moon.    .  .  . 

1  768,  June  29.6,  Mag.  1.21. 
July   1  3.  6,  Total. 
Dec.     8  .  8,  Partial. 
Dec.  23.2,  Mag.  1.75. 
1  769.  Jan.     7  .  6,  Partial. 
June   3.6,  Total. 
June  1  8.  8.  Mag.  i.o4. 
Nov.  27.8,  Total. 
Dec.  1  2.  7,'  Mag.  0.77. 

i8o4,  July  22.2.  Mag.  0.91. 
Aug.     5.  2^  Total. 
Dec.  3  1.  6.  Partial 
i8o5,Jan.  1  4.  8.  Mag.  i  .74. 
Jan.  3o.3j  Partial. 
June  26.6,  Partial. 
July  1  1.  3.  Mag.  1.37. 
July  2  5.  7.  'Partial. 
Dec.  20  .  5,  Total. 
i8o6,Jan.    4.5,  Mag.  0.78. 

1  84o.  Aug.  12.7.  Mag.  o  .  C  i  . 
Aug.  26.6,  Total. 
1  84  1.  Jan.   22.3,  Partial. 
'Feb.     5.6,  Mag.  1.72. 
Feb.   20.9,  Partial. 
July  1  8.  i,  Partial. 
Aug.    1.9,  Mag.  1.67. 
Aug.  16.4,  Partial. 
1842,  Jan.  1  1  .  i,  Annular. 
Jan.  26.2,  Mag.  0.79. 

1876,  Sept.    3.4,  Mag.  o.33. 
Sept-.  17.4,  Total. 
1877.  Feb.   12.8,  Partial. 
'Feb.  27.3,  Mag.  1.62. 
Mar.  1  5.  6,  Partial. 
Aug.    9.7,  Partial. 
Aug.  23.5,  Mag.  1.66. 
Sept.   8  .  o,  Partial. 
1878,  Feb.     1.8,  Total. 
Feb.  17.9,  Mag.  0.82. 

1786.  July  10.9,  Mag.  i.  06. 
July  24  .  9,  Total. 
Dec.  20.2.  Partial. 
1  78  7,  Jan.     3.  5!  Mag.  1.74. 
Jan.   18.9^  Partial. 
June  1  5.  2,  Total. 
JuneSo.i,  Mag.  1.20. 
Dec.    9  .  2,  Total. 
Dec.  24.1,  Mag.  0.77. 

1822,  Au£.    2.5,  Mag.  0.76. 
Aug.  1  6.  5,  Total. 
l823.  Jan.   11.9.  Partial. 
Jan.   26  .  2^  Mag.  1.73. 
Feb.    lo.G,  Partial. 
July     7  .  7.  Partial. 
July  22.6,  Mag.  i.  5i. 
Aug.     6.  i,  Partial. 
Dec.  3  1.  8,  Annular. 
1824.  Jan.    1  5.  8,  Mag.  0.78. 

i858,  Aug.  24.  i,  Mag.  0.47. 
Sept.    7.  i,  Total. 
1859,  Feb.     2  -.5,  Partial. 
Feb.   16.9,  Mag.  i  .69. 
Mar.    4.3.  Partial. 
July  29.4,  Partial. 
Aug.  1  3.  2,  Mag.  i.8l. 
Aug.  27.7,  Partial. 
1860.  Jan.  22.5.  Annular. 
'Feb.     6.  G'.  Mag.  0.81. 

i894;Sept.  1  5.  7,  Mag.  0.21. 
Sept.  29.7,  Total. 
:895.  Feb.  24.  i,  Partial. 
Mar.  11.7,  Mag.  i.56. 
Mar.  26.9,  Partial. 
Aug.  20.0,  Partial. 
Sept.    4-7,  Mag.  i.54. 
Sept.  19  .  3,  Partial. 
1896.  Feb.  1  3.  2,  Total. 
'Feb.  28.4.  Mag.  o.83. 

Sun  
Sun 

Moon  
Sun  

Moon 

Sun  

Moon. 

Moon  
Sun  
Sun  

Sun 

Sun  .  .    . 

Moon     .... 

Sun  
Sun  
Moon  

Moon  
Sun  
Sun  
Moon  
Sun  

Sun 

Moon 

Sun  .  . 

Sun  
Moon  

Moon  
Sun 

Sun  
Moon  
Sun  
Sun  
Moon  

Sun 

Sun  
Moon  

TABLE   VI. — TRANSITS  OF  MERCURY   OVER  THE   SUN  S  DISC. 


Date. 

iJtira- 
tion. 

Date. 

Duru- 
tion. 

Date. 

l/ura- 
tion. 

Date. 

Dura- 
tion. 

h,    m. 

h.   m. 

h.    m. 

h.    m. 

i63i,Nov.  6 

5    23 

1  707,  May  5 

7  54 

1776,  Nov.  2 

I    13 

1  835,  Nov.  7 

5     8 

1  644^  Nov.  8 

3  57 

1710,  Nov.  6 

5  24 

1782,  Nov.  12 

I    15 

1  845,  May  8 

6  45 

1  65  1,  Nov.  2 

3  3i 

1723,  Nov.  9 

4  58 

1  786.  May  3 

5  28 

1  848.  Nov.  9 

5  23 

I66i,May  3 

7  36 

1736,  Nov.  10 

2    42 

1789,  Nov.  5 

4    52 

1  86  1,  Nov.  ii 

4     J 

1  664,  Nov.  4 

5   17 

1  740,  May  2 

3     o 

1  799,  May  7 

7  25 

1  868,  Nov.  4 

3  3i 

1674,  May  6 

4  3o 

1  74  3,  Nov.  4 

4    32 

1  802.  Nov.  8 

5  27 

1878,  May  6 

7  47 

1677,  Nov.  7 

5  i3 

i753,May  5 

7  47 

i8i5jNov.  ij 

4  28 

i88i.Nov.  7 

5   18 

1690,  Nov.  9 

3  36 

i756,Nov.  6 

5  25 

1  822.  Nov.  4 

2    43 

1  89  1,  May  9 

5     8 

1  697,  Nov.  2 

3  56 

1769,  Nov.  9 

4  47 

1  8  32  !  May  5 

6  56 

1894.  Nov.  10 

5  i5 

330 


TABLE   VII. — ASTRONOMICAL   REFRACTIONS. 


App. 
Alti- 
tude. 

Mean 
Refrac- 
tion. 

Differ- 
ence for 
10'. 

tude. 

Mean 
Refrac- 
tion. 

Differ 
ence. 

tude. 

Mean 
Refrac- 
tion. 

Differ 
ence. 

Exter- 
nal 
Therm 

Factor 
T. 

Kxter- 
nal 
Therm 

Factor 
T. 

0      0 
10 
20 

34   54.1 
32   49-2 

3o  52.3 

124-9 
116.9 

8  3o 
4o 
5o 

6    8.4 

6      1.8 

5  55.4 

6.6 

6.4 

6. 

27 
28 
29 

52'.  8 
48.2 
43.8 

4.6 
4.4 

4_ 

—  IO 

9 

8 

I.I30 
I.I28 
I.  125 

42 

43 
44 

I.OlS 
I  .Oil 
I  ,O09 

3o 
4o 

29     3.5 

27  22.7 

100.8 

9     ° 

IO 

5  49-3 
5  43.3 

•  I 

6.0 

c    _ 

3o 
3i 

lu 

.  I 

3.9 

3_ 

7 
6 

I  .  12; 
I  .120 

45 
46 

1  .007 
i  .oo5 

5o 

25  49.8 

92.9 

85.2 

20 

5  37.6 

0.7 
5.6 

32 

32.1 

•7 
3.4 

5 

I.II8 

47 

i.oo3 

I        0 

24  24.6 

.   3o 

5  32.o 

5r 

33 

28.7 

4 

i.n5 

48 

I.OOI 

10 

23       6.7 

77-9 

4o 

5  26.5 

.  0 
c 

34 

25.  ^ 

3.3 

3. 

i.ii3 

49 

i  .000 

20 
3o 
4o 

21  55.6 
20  50.9 
19  51.9 

71.1 
64-7 
5g.o 

CO 

5o 

10      0 

10 

5  2i.3 
5  16.2 

5     II.  2 

0*2 

s!o 

40 

35 
36 
37 

22.' 
I9.3 

i6.5 

.  i 
3.o 

2 

—    I 
0 

i  .no 

I  .  lOfi 

i  .106 

5o 
5i 

52 

.998 
.996 
.994 

5o 

18  58.o 

oJ.g 
49.4 

20 

5     6.4 

.0 

4.7 

38 

i3.8 

«7 
.6 

+     I 

i  .io3 

53 

•992 

2       O 

10 

20 

3o 

4o 

18     8.6 

17    23.0 

i  6  40.7 
16     0.9 
i5  23.4 

45.6 

42.3 

39.8 
37.5 

oc    c 

3o 
4o 
5o 

II       0 
IO 

5     1.7 
4  57.2 
4  52.8 
4  48.5 
4  44.3 

4.5 
4.4 
4.3 
4.2 

4T 

4i 

42 
43 

II  .2 

8.7 

6.3 
4.o 

1.8 

.5 
.4 
.3 

.2 

2 

3 
4 
5 
6 

I  .101 

i  .09!: 
i  .096 
i  .094 
1.091 

54 
55 
56 

57 

58 

.990 

.988 
.986 
.984 
.982 

5o 

i4  47.8 

OO  •  0 

33.2 

20 

4  4o.2 

•  I 

3.9 

44 

59.7 

.  I 
.O 

7 

i  .089 

59 

.980! 

3     o 

i4  i4.6 

3o  o 

3o 

4  36.3 

•5     „ 

45 

57.7 

8 

1.087 

60 

.978 

IO 

i3  43.7 

_Q       _- 

4o 

4  32.4 

3? 

46 

55.7 

•  O 

9 

i.o84 

61 

•977 

20 

i3  i5.o 

28.7 

5o 

4  28.7 

0.7 

0      _ 

47 

53.8 

•9 

10 

i  .082 

62 

.975 

3o 

12  48.3 

26.7 

12       0 

4  25.o 

0.7 

0     fi 

48 

5i.9 

•9 

n 

1.080 

63 

.973 

4o 

12    23-7 

24.6 

Oo     ._. 

10 

4  21.4 

>3  .  O 

•J     / 

49 

50.2 

•  7 

12 

1.078 

64 

.971 

5o 

12       0.7 

2o  .  O 

21.8 

2O 

4  18.0 

0.4 

3.4 

5o 

48.4 

•  7 

13 

i.o75 

65 

.969 

4     o 

ii  38.g 

20  .  6 

3o 

4  i4.6 

3  3 

5i 

46.  7 

.6 

i4 

i.o73 

66 

.967 

IO 

ii   i8.3 

4o 

4  ii.  3 

30 

52 

45.i 

i5 

1.071 

67 

.965 

20 

10  58.6 

19.7 

5o 

4    8.0 

•  o 

3_ 

53 

43.5 

16 

1.069 

68 

.964 

3o 
4o 

10  39.6 

IO    21  .2 

19.0 

18.4 

i3     o 

10 

4    4.9 

4     1.8 

.  i 
3.i 

o    n 

54 
55 

4i.g 
40.2 

.5 

17 

18 

1.066 
i.o64 

69 

70 

.962 
.960 

5o 

10     3.3 

17.9 
16.8 

20 

3  58.8 

O  .  O 

2.9 

56 

38.9 

.4 

'9 

1.062 

.958 

5     o 

9  46.5 

-EC 

3o 

3  55.9 

57 

37.5 

20 

i  .060 

72 

.956 

10 

9  3o.9 

I  3  •  0 

4o 

3  53.0 

'I 

58 

36.i 

21 

i.o57 

73 

.955 

20 

9  16.0 

14.9 

5o 

3  5o.2 

.0 

.8 

59 

34.7 

/ 

22 

i.o55 

74 

.953 

3o 

9     i-9 

14.1 

y  Q         C 

i4     o 

3  47-4 

60 

33.3 

.4 
.  3 

23 

i.o53 

75 

.95i 

4o 

8  48.4 

I  *J  .  3 

128 

10 

3  44.7 

•7 
.6 

61 

32.0 

.3 

24 

i.oSi 

76 

.949 

5o 

8  35.6 

i2'.3 

20 

3  42.1 

.6 

62 

3o.7 

.3 

25 

1.049 

77 

.948 

6     o 

8  23.3 

3o 

3  39.5 

c 

63 

29.4 

26 

1.047 

78 

•  946 

10 

8  ii.  6 

11.7 

T    T          *3 

4o 

3  37.o 

.  3 
c 

64 

28.2 

•  2 

0 

27 

i.o44 

79 

.944 

20 

8     o.3 

T  I  •  O 

i  o.  8 

5o 

3  34.5 

.  3 
•>    A 

65 

26.9 

•  J 

28 

i  .042 

80 

.942 

3o 
4o 

7  49.5 
7  39.2 

io.3 

5     o 
6     o 

3  32.i 
3  18.6 

2  .4 

i3.5 

66 
67 

25.7 
24.5 

.  2 

.2 

29 

3o 

.o4o 
.o38 

81 
82 

.94i 
.939 

5o 

7  29.2 

I  O  .  O 

9.5 

7     o 

3     6.6 

2  •  O 

10.8 

68 

23.3 

.  2 

.1 

3i 

.o36 

83 

.937 

7     ° 

IO 

7  19.7 
7  io.5 

9.2 

8Q 

8     o 

2  55.8 
2  46.i 

9-7 

80 

69 
70 

22.2 
21.0 

.2 

32 

33 

.o34 

.032 

84 
85 

.935 
.934 

20 

7     1-7 

.0 
81 

20      0 

2  37.3 

.  o 
8_ 

7* 

19.9 

*X 

34 

.o3o 

86 

.932 

3o 

6  53.3 

•  4 
8_ 

21       0 

2    29.3 

.  O 
_     / 

72 

18.8 

•  I 

35 

.028 

87 

.93o 

4o 

6  45.1 

.  2 

22       O 

2    21.9 

7-4 

6_ 

73 

17.7 

.  I 

36 

.026 

88 

.929 

5o 

6  37.2 

7-9 
7-6 

23       0 

2    l5.2 

•7 

6.3 

74 

16.6 

•  I 

.1 

37 

.024 

89 

.927 

8     o 

10 

6  29.6 

6    22.3 

7.3 

24       0 
25       0 

2       8.9 
2        3.2 

M 

75 
80 

i5.5 

10.2 

5.3 

38 
39 

.022 
.019 

90 
91 

.925 

.924 

20 

6  i5.2 

7  •  * 

60 

26       0 

I     57.8 

J  .  4 
c    n 

85 

5.  i 

c    T 

4o 

.017 

92 

.922 

3o 

684 

•  o 

27     o 

i   52.8 

D  .  O 

9° 

0.0 

O.I 

4i 

.oi5 

93 

.920 

TABLES  VII.,  VIII.,  AND  IX. 


331 


TABLE  VII. — ASTRONOMICAL   REFRACTIONS. 


Barom. 

Factor  B. 

Barom. 

Factor  B. 

Barom. 

Factor  B. 

Attached 
Therm. 

Factor  t. 

Attached 
Therm. 

Factor  t  . 

inches. 

inches. 

inches. 

0 

27.9 

0.943 

29.0 

0.980 

3o.I 

.017 

—  i5 

.oo4 

4o 

0.999 

28.0 

•  946 

29.1 

.983 

30.2 

.020 

10 

.oo4 

45 

0.999 

28.1 

.949 

29.2 

.987 

3o.3 

.024 

.  5 

.oo3 

5o 

0.998 

28.2 

.953 

29.3 

.990 

3o.4 

.027 

o 

.oo3 

55 

0.998 

z8.3 

•  956 

29.4 

.993 

3o.5 

.o3i 

+    5 

.oo3 

60 

0.997 

28.4 

.960 

29.5 

•997 

3o.6 

.o34 

10 

.002 

65 

0.997 

28.5 

.963 

29.6 

.000 

3o.7 

.o37 

:5 

.002 

7° 

0.997 

28.6 

.966 

29.7 

.oo3 

3o.8 

.o4i 

20 

.op  i 

75 

0.996 

28.7 

.970 

29.8 

.007 

3o.g 

.o44 

25 

.001 

80 

0.996 

28.8 

.973 

29.9 

.010 

3i.o 

.047 

3o 

.000 

85 

0.995 

28.9 

.976 

3o.o 

.014 

3i.i 

.o5o 

35 

.000 

90 

0.995 

TABLE  VIII. — REDUCTION  OF  THE   MOON'S   EQUATORIAL   PARALLAX. 


tude. 

Moon's  Equatorial 
Parallax. 

Lati- 
tude. 

Moon's  Equatorial 
Parallax. 

Lati- 
tude. 

Moon's  Equatorial 
Parallax. 

53' 

57' 

61' 

53' 

57' 

61' 

53'   |  57' 

61' 

2 

0.0 

o.  o 

o.o 

32 

3.0 

3.2 

3.4 

62 

8.3 

8.9 

9.5 

4 

0.  I 

O.  I 

0.  I 

34 

3.3 

3.6 

3.8 

64 

8.6 

9.2 

9.9 

6 

0.  I 

O.I 

0.  I 

36 

3.6 

3.9 

4.2 

66 

8.9 

9.5 

10.2 

8 

0.2 

O.2 

0.2 

38 

4.0 

4.3 

4.6 

68 

9.1 

9.8 

io.5 

10 

o.3 

o.3 

0.4 

4o 

4.4 

4-7 

5.o 

7° 

9-4 

10.  I 

10.8 

12 

o.5 

o.5 

o.5 

42 

4-7 

5.i 

5.4 

72 

9.6 

io.3 

ii  .  i 

14 

0.6 

0.7 

0.7 

44 

5.i 

5.5 

5.9 

74 

9.8 

10.6 

ii.  3 

16 

0.8 

0.9 

0.9 

46 

5.5 

5.9 

6.3 

76 

10.  0 

10.8 

ii.5 

18 

I  .0 

1  .1 

I  .2 

48 

5.8 

6.3 

6.7 

78 

10.2 

10.9 

11.7 

20 

1.2 

i.3 

i.4 

5o 

6.2 

6.7 

7-1 

80 

io.3 

n.  i 

1-1.9 

22, 

i.5 

1.6 

x-7 

52 

6.6 

7-i 

7-6. 

82 

10.4 

II  .2 

12.  0 

24 

J-7 

1.9 

2.0 

54 

6.9 

7.5 

8.0 

84 

io.5 

ii.  3 

12.  I 

26 

2.0 

2.2 

2.3 

56 

7.3 

7.8 

8.4 

86 

10.6 

n.  4 

12.2 

28 

2.3 

2.5 

2.7 

58 

7.6 

8.2 

8.8 

88 

10.6 

ii.  4 

12.2 

3o 

2.6 

2.8 

3.o 

60 

8.0 

8.6 

9.2 

90 

10.6 

11.  4 

12.2 

TABLE   IX. — ELEMENTS   OF   PERIODICAL  COMETS. 


JT                i  Time  of  Perihelion 

Longitude 
of  Peri- 
helion. 

Ascend- 
ing Node. 

Inclina- 
tion of 
Orbit. 

Eccen- 
tricity. 

Semi-ma- 
jor Axis. 

Period 
in 
Yenre. 

Mo- 
tion. 

Hallev's....  i835.  Nov.  16 

3o4  32 

55    10 

17  45 

0.9674 

17.9879 

76.291 

R. 

Tattle's  .... 
Faye's  
Biela's  

D'Arrest's  . 
Brorsen's... 
Winnecke's 

1871,  Dec.  2 
1866,  Feb.  i4 
1  852,  Sept.  23 
i857,  Nov.  29 
1  868,  April  18 
1869,  June  3o 

116     4 

49    S? 

109     8 
323     3 
116     2 

275  55 

269    I7 
209    42 

245  5i 
i48  27 
101   i4 
n3  3a 

54  17 

II     22 

12   33 
i3  56 
29  23 
10  48 

.8209 

.5575 
.7559 
.6598 
.8080 
.7619 

5.7266 
3.  8018 
3.6262 
3.44o4 
3.1096 
3.i499 

13.702 
7-4i3 
6.619 
6.38i 

5.484 
5.59i 

D. 
D. 
D. 
D. 
D. 
D. 

Encke's  .... 

1871,  Dec.  29 

i58   12 

334  34 

i3     7 

.8493 

2.2097 

3.285 

D.j 

332 


TABLES  X.,  XI.,  AND  XII. 


TABLE  X. — ALTITUDES   OF  THE   PRINCIPAL   LUNAR   MOUNTAINS. 


Name. 

Altitude 
in  Feet. 

stenographic 
Position. 

Name. 

Altitude 
in  Feet. 

Stenographic 

Longi- 
tude. 

Latitude. 

Longi- 
tude. 

Latitude. 

Newton  .... 

28800 

16  E. 

77  S. 

Tycho  

17800 

12    E. 

43  S. 

Curtius  

22800 

3  W. 

67  S. 

Kircher  .... 

17600 

43  E. 

67  S. 

Casatus  .... 

20800 

35  E. 

74  S. 

Pythagoras  . 

16900 

60  E. 

63  N. 

Calippus  .  .  . 

20400 

10  W. 

39  N. 

Clavius  

16800 

i5  E. 

58  S. 

Posidonius  .  . 

19800 

29  W. 

3i  N. 

Endymion  .  . 

16700 

55  W. 

53  N. 

Short  

18700 

10  E. 

74  S. 

Catharina  .  . 

i64oo 

23  W. 

17  S. 

Moretus  .... 

i84oo 

7  E. 

70  S. 

Theophilus  . 

16900 

26  W. 

ii   S. 

Mutus  

i83oo 

3o  W. 

63  S. 

Harpalus  .  .  . 

i58oo 

'  44  E. 

53  N. 

Huyghens  .  . 

18000 

2    E. 

20    N. 

Eratosthenes 

16600 

ii   E. 

i4  N. 

Blancanus  .  . 

18000 

21    E. 

63  S. 

Werner  .... 

i56oo 

3  W. 

28  S. 

TABLE  XL — DIAMETERS  OF  SOME   OF  THE  ANNULAR  MOUNTAINS 
ON  THE   MOON. 


Name. 

Diameter 
in  Knglish 
Miles. 

Selenographic 
Position. 

Name. 

Diameter 
in  Knglish 
Mile*. 

(-1  ll-llo 

Pofi 

graphic 
ion. 

Longi- 

Latitude. 

Longi- 
tude. 

Latitude. 

Clavius  

143 

15   E. 

58  S. 

Scheiner  .  .  . 

70 

26  E. 

60  S. 

Schikard  .  .  . 

I  34 

55  E. 

44  S. 

Posidonius.  . 

62 

29  W. 

3l    N. 

Ptolemy  .... 

ii5 

3  E. 

9  S. 

Plato  

60 

9  E. 

5i  N. 

Schiller  .... 

n3 

38  E. 

62  S. 

Flamsteed  .  . 

60 

44  E. 

5  S. 

Gauss    

i  ii 

75  W. 

S?  N. 

Piccolomini  . 

58 

3  1  W. 

29  S. 

Riccioli  .... 

1  06 

75  E. 

2    S. 

Copernicus.  . 

55 

20    E. 

9N. 

Hipparchus  . 

97 

5  W. 

6  S. 

Fabricius  .  .  . 

55 

4  1  W. 

42    S. 

Boussinpault 

92 

55  W. 

68  S. 

Tycho  

54 

12    E. 

43  S. 

Cleomedes  .  . 

78 

55  W. 

27  N. 

Aristarchus  . 

28 

47  E. 

23  N. 

Hevelius  .  .  . 

70 

67  E. 

2    N. 

Kepler  

22 

38  E. 

8  N. 

TABLE  XII.— TRANSITS  OF  VENUS  OVER  THE   SUN'S  DISC. 


Year. 

Greenwich  Mean  Time  of 
Conjunction. 

Duration  of  Transit. 

Least  Distance  of 
Venus  from  Sun's 
Centre. 

h.       m.        e. 

h.      m.        s. 

/        // 

i639 

Dec.    4,      6       o     20 

6     34       o 

9       o  S. 

I76l 

June  5,    17     35     i4 

6160 

9     3o  S. 

1769 

June  3.      9     58     34 

5     Sg     46 

10     10  N. 

1874 

Dec.  8,    1  6       8     24 

4      9    22 

i3     5i  N. 

1882 

Dec.   6,      4     1  6     24 

6       3     26 

10     29  S. 

2004 

June  7,    20     5i     24 

5    29    4o 

ii      19  S. 

2012 

June  5.    1  3     17     4o 

6     4i     3o 

8     20  N. 

TABLES  XIII.,  XIV.,  AND  XV. 


333 


TABLE  xiii. — SCHWABE'S  OBSERVATIONS  OF  THE  SOLAR  SPOTS. 


Year. 

observing 
Day.. 

Uroupa  of 
Spots  ob- 
served. 

Days  DU  which 
the  Sun  was 
free  fi-om  Spots. 

Year. 

Number  of 
observing 
-  Days. 

Groups  ot 
Spot»  ob- 
served. 

Days  on  which 
the  Sun  was 
free  from  Spots 

itiab 

277 

I  l8 

22 

1  846 

3i4 

i57 

I 

1827 
1828 

273 
282 

161 
226  max. 

2 
O 

1847 

1  848 

276 
278 

257 
33o  max. 

O 
o 

1829 

244 

199 

O 

i849 

285 

238 

0 

i83o 

2I7 

190 

I 

i85o 

3o8 

1  86 

2 

i83i 

239 

i4g 

3 

i85i 

3o8 

i5i 

0 

i832 

270 

84 

4g 

1862 

337 

125 

2 

i833 

267 

33  min. 

i39 

i853 

299 

91 

3 

i834 

2?3 

5i 

120 

i854 

334 

67 

65 

i835 

244 

i73 

18 

i855 

3i3 

79 

i46 

iS36 

200 

272 

o 

i856 

321 

34  min. 

i93 

i837 

168 

333  max. 

O 

1867 

324 

98 

52 

i838 

202 

282 

o 

i858 

335 

188 

o 

i839 

205 

162 

o 

i859 

343 

205 

0 

i84o 

263 

i5a 

3 

1860 

332 

2  10  max. 

0 

i84i 

283 

102 

i5 

1861 

322 

204 

0 

1842 

3o7 

68 

64 

1862 

3i7 

1  60 

3 

i843 

324 

34  min. 

149 

i863 

33o 

124 

2 

1  844 

321 

52 

1  1  1 

1  864 

325 

i3o 

4 

i845 

332 

n4 

29 

i865 

3o7 

93 

26 

TABLE   XIV. — PARALLAX  OF  FIXED  STARS. 


Star. 

Mag- 
nitude. 

Parallax. 

Distance 
(Sun's  Dis- 
tance^). 

Light 
Interval  in 
Years. 

Name  of  Observer. 

a  Centauri 

I 

0.928 

222268 

3.5 

Henderson  and  Moesta. 

6  1  Cygni 

6 

0.553 

372992 

5.9 

Struve  and  Auwers. 

2ii85  Lalande 

7 

o.5oi 

/                 r 
4I170D 

6.5 

Winnecke. 

/3  Centauri 

i 

0.470 

43886i 

6.9 

Maclear. 

(i  Cassiopeae 

5 

0.342 

6o3n3 

9.5 

Struve. 

34  Groombridge 

8 

0.307 

671872 

10.6 

Auwers. 

Capella 

i 

o.3o5 

676278 

10.7 

Struve. 

21258  Lalande 

8 

0.271 

761125 

12.0 

Kriiger. 

I74i5  Oeltzen 

8 

0.247 

835o8o 

l3.2 

Kruger. 

a  Draconis 

5 

0.246 

838475 

13.2 

Brunnow. 

Sinus 

i 

o.ioS 

1068730 

16.0 

Maclear. 

TABLE   XV. — ELEMENTS  OF  THE   ORBITS  OF   BINARY  STARS. 


Name  of  Star. 

A.  R. 

Dec. 

Semi-ma- 
jor Axis. 

Kccen- 
tricity. 

Period  of 
He  volution 
in  Years. 

By  whom 
computed. 

f  Herculis 

248  54 

+3i   52 

1.25 

0.448 

36 

Villarceau. 

t  Cancri 

120    54 

+  18     5 

0.89 

0.443 

58 

Madler. 

K  Ursa;  Majoris 
TI  Coronas 

167  32 
229  i5 

+  32    22 

+3o  49 

2.29 
I  .20 

o.4o3 
0.4o4 

61 
67 

Madler. 
Villarceau. 

a  Centauri 

217    22 

—60    12 

i5.5o 

o.gSo 

80 

Jacob. 

T  Ophiuchi 

268  43 

—  8  10 

0.82 

o.o37 

87 

Madler. 

70  Ophiuchi 

269  28 

+    2    32 

4.5o 

o.48o 

92 

Madler. 

X  Ophiuchi 

245  5o 

4-  2   18 

o.84 

o.477 

95 

Hind. 

£  Libra 

239     i 

—  10  57 

1.29 

0.000 

io5 

Madler. 

32io  CassiopeiE 

359  36 

+57  36 

I  .00 

o.575 

i46 

Madler. 

£  Bootis 

221        7 

+  19  43 

5.59 

0.454 

1  60 

Madler. 

y  Virginia 

1  88  3i 

-  o  37 

3.86 

0.880 

169 

Madler. 

S  Cygni 

295     4  '  +44  45 

I.8I 

c  606 

n8 

Hind. 

334: 


TABLE   XVI. 


TABLE   XVI. — VARIABLE   STARS. 


No. 

Star. 

K.  A. 

18TO. 

Declination. 
1870. 

I'eriod. 
Days. 

S§§!  I    D>— 

h.     m.     a. 

O             I 

I 

oCeti 

2    12   4? 

—  3  34.i 

331.34 

2       tO  12 

Fabricius,  i596 

2 

ft  Persei 

2    59   43 

+4o  27.2 

2.867 

2.5"     4 

Montanari,  1669 

3 

t  Auriga? 

4    52    38 

+43  37.7 

35o 

3.5  <   4.5 

Heis,  1  846 

4 

R  Leonis 

9  4o  34 

+  12       1.8 

3l2 

5      <ii.5 

Koch,  1782 

5 

rt  Argtis 

IO   4o       2 

-59     o.i 

67  years 

i       •'   4 

Burchell,  1827 

C 

R  Hydras 

l3    22    37 

—22    36.4 

449 

4      <io 

Maraldi,  1704 

7 

3o  Herculis 

l6    24    22 

+42    10.  1 

106 

5      '   6 

Baxendell,  1857 

8 

Nova  Ophiuchi 

16  52  i3 

—12  4i.4 

4.5  <i3.5 

Hind,  1  848 

9 

K  Coronse  Aust. 

18  24  24 

-38  48.9 

3       '    6 

Halley,  1676 

10 

R  Scuti  Sobies. 

18  4o  33 

—  5  5o.5 

7i.75 

5       <   9 

Pigott,  1795 

1  1 

ft  Lyra* 

18  45  17 

+33  12.7 

12.91 

3.5  '    4.5 

Goodricke,  1784 

12 

*Cygni 

19  45  34 

+  32  35.2 

4o6 

5       <i3 

Kirch,  1687 

i3 

T]  Aquilaj 

19  45  5i 

+  o  4o.4 

7.176 

3.5  '   4.5 

Pigott,  1784 

i4 

34  Cygni 

20    l3       0 

+37  37.8 

1  8  years 

3       '    6 

Jansen,  1600 

i5 

24  Cephei 

20    23    4l 

+88  44.o 

73  years 

5       <n 

Pogson,  1  8  56 

16 

ft  Cephei 

21  3g  3i 

+58  n.  i 

5  or  6  y. 

4       '-    6 

W.Herschel,i782 

i? 

d  Cephei 

22    24    21 

+57  45.o 

5.366 

3.7"    4-8 

Goodricke,  1  784 

TABLE   XVII. — MEASURED  ARCS  OF  A  MERIDIAN  OF  THE   EARTH. 


Country. 

latitude  of 
Middle  of  Arc. 

Arc  Measured. 

length  in 
Eng  i-h  Feet. 

Length  in 
English  Miles. 

Russia 
England 
France 
United  States 
India 
Peru 
South  Africa 

0         /            II 

5745     7.0 

55  21  36.2 
44  5i    2.2 
42  58  52.5 
18  5o    9.7 
—  i  3i     o.o 

—  32       2  42.0 

0           1              II 

25  20     8.5 
10  56     4.6 

12    22    12.7 
3    22    39.2 
21    21     17.2 

3     7     3.5 
4  36  48.6 

9,  258,  006 
3;994,i83 
4,509,402 
1,231,077 
7,755,827 
i,i3i,o57 
1,678,374 

I753.4l 

756.47 
854.13 
233.i6 
i468.9i 
214.22 
3i7.87 

EXPLANATION  OF  THE  TABLES. 


Table  I.,  page  321,  contains  the  principal  elements  of  the  plan- 
etary  system,  with  the  exception  of  the  minor  planets.  These 
elements  have  been  taken  from  Le  Verrier's  Annales  de  1'Observa- 
toire,  tome  second,  p.  58-61,  as  far  as  they  are  there  given ;  other 
numbers  depending  upon  these  have  been  derived  from  them  by  • 
computation ;  and  the  remainder  of  the  Table  has  been  derived 
from  various  sources,  but  chiefly  from  Hind. 

Table  II.,  pages  322  and  323,  contains  the  elements  of  the  minor 
planets.  These  elements  have  been  derived  from  the  Berlin  As- 
tronomisches  Jahrbuch  for  1866,  with  the  exception  of  the  last 
three,  which  were  derived  from  recent  periodicals. 

Table  III.,  page  324,  contains  the  elements  of  the  satellites  of 
the  primary  planets.  These  elements  were  derived  from  a  com- 
parison of  various  authorities,  such  as  Herschel's  Astronomy, 
Chambers's  Hand-book  of  Astronomy,  Hind's  Solar  System,  and 
Chazallon's  Annuaire  des  Marees  pour  1860. 

Table  IV.,  pages  325  and  326,  contains  a  catalogue  of  all  the 
eclipses  of  the  sun  that  will  be  visible  in  the  city  of  Boston  from 
1865  to  1900.  It  is  copied  from  the  American  Almanac  for 
1831,  and  was  computed  by  Mr.  R.  T.  Paine. 

Table  V.,  page  327,  contains  a  catalogue  of  eclipses  designed  to 
illustrate  several  important  principles.  It  shows,  first,  that  seven 
eclipses  may  occur  in  one  year ;  and,  second,  it  illustrates  the  prin- 
ciple of  the  Saros.  The  data  for  the  past  eclipses  were  derived 
from  the  English  Nautical  Almanac ;  and  those  for  future  eclipses 
were  derived  chiefly  from  Chambers's  Hand-book  of  Astron- 
omy. 


336  ASTRONOMY. 

Table  VI.,  page  327,  contains  a  complete  catalogue  of  the  tran- 
sits of  Mercury  over  the  sun's  disc  from  1631  (the  first  transit  ob- 
served) to  the  close  of  the  present  century.  It  is  derived  from 
Delarnbre's  Astronomic,  t  ii.,  p.  518. 

Table  VlL,  pages  328  and  329,  contains  Bessel's  Astronomical 
Refractions  in  an  abridged  form.  It  requires,  in  addition  to  the 
observed  apparent  altitude,  an  observation  of  the  height  of  the 
barometer,  upon  which  depends  the  factor  B ;  of  the  thermometer 
attached  to  the  barometer,  upon  which  depends  the  factor  t ;  and 
of  the  temperature  of  the  external  air,  upon  which  depends  the 
factor  T. 

Take  the  mean  refraction  corresponding  to  the  observed  alti- 
tude ;  take  the  factor  B  corresponding  to  the  height  of  the  ba- 
rometer ;  also  the  factor  t  corresponding  to  the  attached  thermom- 
eter; and  the  factor  T  corresponding  to  the  external  thermom- 
eter. Multiply  these  four  numbers  together,  and  the  product  will 
be  the  true  refraction. 

Example.  The  observed  apparent  altitude  of  a  star  was  34°  11' 
15" ;  the  barometer,  28.856  inches ;  the  external  and  the  attached 
thermometers  both  stood  at  +  19°.  6  Fahr.  It  is  required  to  com- 
pute the  refraction. 

Mean  refraction  for  34°  11'  15" 1'  24".8. 

Barometer,  28.856.      Factor  B,  0.975. 
i  no  „  (  Factor  t.   1.001. 
Thermometer'19'6  I  Factor  T,  1.061. 
Product,        0.975x1.001x1.061  =1.0355. 

True  refraction^  84".8  x  1.0355  =  1'  27//.8. 
For  small  altitudes,  when  great  accuracy  is  required,  the  com- 
putation is  most  conveniently  performed  by  logarithms.     A  Ta- 
ble, which  furnishes  the  logarithms  of  all  these  factors,  is  given 
in  my  Practical  Astronomy,  pages  364-5. 

Table  VIII.,  page  329,  shows  the  quantity  by  which  the  moon's 
equatorial  horizontal  parallax  must  be  diminished,  to  obtain  the 
horizontal  parallax  belonging  to  any  other  latitude.  This  reduc- 
tion is  given  for  three  values  of  the  moon's  equatorial  parallax, 
viz.,  53',  57',  and  61' ;  and  for  any  other  value,  the  equatorial  par- 
allax may  be  easily  found  by  interpolation. 


EXPLANATION  OF  THE  TABLES.  337 

Table  IX.,  page  329,  contains  the  elements  of  the  seven  comets 
Whose  periods  have  been  well  established.  These  elements  have 
been  derived  chiefly  from  the  Astronomische  Nachrichten. 

Table  X.,  page  330,  exhibits  the  altitude  in  English  feet  of  the 
principal  lunar  mountains  according  to  the  observations  of  Beer 
pnd  Madler. 

Table  XL,  page  330,  exhibits  the  breadths  in  English  miles  of 
some  of  the  larger  craters,  or  annular  mountains  on  the  moon's 
surface,  according  to  the  observations  of  Beer  and  Madler. 

Table  XII.,  page  330,  contains  a  catalogue  of  all  the  transits 
,  of  Yenus  over  the  sun's  disc  from  1639  (the  first  ever  observed) 
to  the  end  of  the  21st  century.    It  is  derived  from  Delambre's 
Astronomic,  t  ii.,  p.  473. 

Table  XIII.,  page  331,  exhibits  the  results  of  39  years  of  ob- 
servations of  the  solar  spots  by  M.  Schwabe,  of  Dessau,  in  Ger- 
many. Column  2  shows  the  number  of  days  in  each  year  upon 
which  observations  were  made ;  column  3  shows  the  number  of 
groups  of  spots  observed ;  and  column  4  shows  the  number  of 
days  on  each  year  upon  which  the  sun  was  free  from  spots.  These 
observations  decidedly  indicate  a  periodicity  in  the  number  of  the 
solar  spots,  a  maximum  recurring  at  an  interval  of  from  9  to  12 
years. 

Table  XIV.,  page  331,  exhibits  the  results  of  the  best  observa- 
tions hitherto  made  for  determining  the  parallax  of  some  of  the 
fixed  stars.  Several  of  the  results  here  given  are  the  averages  of 
the  determinations  by  two  or  more  astronomers.  The  results  for 
the  two  stars  first  mentioned  are  entitled  to  considerable  confi- 
dence ;  all  the  others  are  to  be  regarded  as  quite  doubtful,  except 
as  indicating  that  the  parallax  can  not  much  exceed  the  quantities 
here  given. 

Table  XV.,  page  331,  furnishes  the  elements  of  those  binary 
stars  whose  periods  are  less  than  two  centuries. 


EXPLANATION  OF  THE  PLATES. 


Plate  I.  is  a  chart  of  the  world  with  cotidal  lines  marked  upon 
it.  The  numerals  upon  the  cotidal  lines  denote  the  hour,  in 
Greenwich  time,  of  high  water  on  the  day  of  new  moon  or  full 
moon.  The  map  is  mainly  copied  from  Professor  Airy's  chart  in 
the  Encyclopasdia  Metropolitana,  Article  Tides,  with  modifications 
suggested  by  the  observations  of  the  United  States  Coast  Survey, 
and  other  recent  observations  in  the  Pacific  Ocean. 

Plate  II.,  Fig.  2,  is  a  representation  of  the  appearance  of  the 
full  moon,  copied  from  the  engraving  of  Beer  and  Madler,  modi- 
fied according  to  a  photographic  picture  taken  at  the  Cambridge 
(Massachusetts)  Observatory. 

Fig.  1  is  a  representation  of  a  small  portion  of  the  moon's  sur- 
face as  seen  with  a  powerful  telescope  near  the  time  of  first  quar- 
ter. This  figure  is  derived  from  MitchePs  Sidereal  Messenger, 
vol.  i.,  p.  32. 

Plate  III.,  Fig.  1,  is  a  representation  of  the  total  solar  eclipse  of 
July  18th,  1860,  as  observed  in  the  northern  part  of  Spain.  The 
figure  is  copied  from  a  photograph  taken  by  De  la  Eue  one 
minute  after  total  obscuration.  Fig.  2  is  copied  from  a  photo- 
graph taken  immediately  previous  to  the  reappearance  of  the  sun. 
In  Fig.  1  the  luminous  protuberances  are  almost  entirely  on  the 
left-hand  side  of  the  sun's  disc,  while  the  right  side  is  almost  en- 
tirely free  from  them.  In  Fig.  2  protuberances  had  come  into 
view  on  the  right-hand  side,  while  those  on  the  left  hand  have 
mostly  disappeared,  showing  conclusively  that  these  protuber- 
ances are  attached  to  the  disc  of  the  sun,  and  not  to  that  of  the 
moon. 

Plate  IV.  contains  representations  of  the  planets  Venus,  Mars, 
Jupiter,  and  Saturn.  The  figure  of  Venus  is  copied  from  a  draw- 


340  ASTRONOMY. 

ing  by  Sehroter,  representing  the  planet  near  its  inferior  conjunc- 
tion. The  figure  of  Mars  is  copied  from  a  drawing  by  Secchi, 
published  with  the  Observations  of  the  Eoraan  Observatory  for 
1856.  The  figure  of  Jupiter  is  copied  from  a  drawing  in  the  Si- 
dereal Messenger,  vol.  i.,  p.  72  ;  and  the  figure  of  Saturn  is  copied 
from  a  drawing  by  Dawes  in  the  Astronomische  Nachrichten,  vol. 
xxxv.,  p.  395.  (See  Frontispiece.) 

Plate  V.  contains  representations  of  several  comets.  Fig.  1  is 
Encke's  comet,  from  a  drawing  by  Struve  in  1828 ;  Fig.  2  is  a 
representation  of  the  head  of  Halley's  comet  as  observed  in.  Oc- 
tober, 1835,  by  Bessel,  showing  the  luminous  jets  which  emanated 
from  the  nucleus ;  Fig.  3  is  a  representation  of  Biela's  comet  as 
observed  in  February,  1846,  by  Struve,  showing  the  division  into 
two  comets ;  Fig.  4  is  a  representation  of  the  great  comet  of  18-43 
as  seen  by  the  naked  eye ;  and  Fig.  5  is  a  representation  of  the 
remarkable  comet  of  1744  as  seen  March  8th,  at  Geneva,  by  Che- 
seaux. 

Plate  VI.  also  contains  representations  of  comets.  Fig.  2  is  a 
representation  of  Donati's  comet  as  it  appeared  to  the  naked  eye 
October  10,  1858,  according  to  a  drawing  by  Professor  Bond ; 
Fig.  1  is  a  representation  of  Halley's  comet  as  it  appeared  to  the 
naked  eye  October  29, 1835,  according  to  Struve;  and  Fig. 3  is  a 
telescopic  view  of  the  head  of  Donati's  comet  as  it  appeared  Oc- 
tober 2, 1858,  according  to  a  drawing  by  Professor  Bond. 

Plate  VII.,  Fig.  1,  is  a  representation  of  the  great  nebula  in 
Andromeda,  copied  from  Herschel's  Astronomy,  Plate  II. ;  Fig.  2 
is  a  representation  of  the  great  cluster  w  Centauri,  copied  from 
Herschel's  Cape  of  Good  Hope  Observations,  Plate  V. ;  and  Fig. 
3  is  a  representation  of  the  great  nebula  in  Orion,  copied  from 
Herschel. 

Plate  VIII. ,  Fig.  1,  is  a  representation  of  the  dumb-bell  nebula ; 
Fig.  3,  the  annular  nebula  in  Lyra ;  Fig.  2,  the  crab  nebula ;  and 
Fig.  4,  the  spiral  nebula — all  copied  from  figures  in  Nichol's  S\*s- 
tem  of  the  World. 


INDEX. 


Page)  POK« 

Aberration,  amount  of. 105  Central  force  inversely  as  square  of 

'           illustrated 105      distance ." 132 

'           of  a"  star  at  the  pole 106 -Central  force  of  a  moving  body 130 

'           of  a  star  in  ecliptic 106'Central  forces,  how  they  vary 131 

Aerolites  described 282  Centrifugal  force,  and  force  of  gravity  23 

orbits  of. 284          "             "      and  form  of  body...  24 

wmr,  r^f                                                                 "      at  any  latitude 24 

"      computed 23 

"      effects  of. 22 

Change  of  seasons,  cause  of. 62 

"       made  greater 63 

Chronometers,  longitude  by 179 


orbits  of. 284 

origin  of. 283 

Alt  tude  and  azimuth  defined 12 

and  azimuth  instrument 41 

instrument  adjusted 4fr 

of  a  body  determined 4( 

Altitudes  measured  by  sextant 5( 

Animal  life  on  the  moon 122 

Annual  equation 15< 

"       parallax  of  stars 292 

Annular  mountains  of  moon 330 

Anomaly,  mean,  and  true 107 

Apparent  motion  of  inferior  planet....  201 
"      of  superior  planet...  202 

Arc  of  meridian,  how  measured 2;' 

Asteroid  system,  origin  of 226 

Asteroids,  brightness  of. 225 

"         distance  of. 225 

"         number  known 225 

"         total  number  of 225 

"        within  orbit  of  Mercury....  22G 

Astronomical  refractions,  table  of 328 

Astronomv  defined 9 


Atmosphere,  illuminating  effect  of.... 

Atmospheric  refraction,  law  of. 

Attractions  of  sun  and  earth 


f>8 


li.'J 


Baily's  beads  described 17G 

Biek's  comet  divided 273 

"         "      history 272 

Binary  star,  mass  of 303 

Binary  stars  defined 299 

"         "     elements  of. 331 

"         "     number  of 302 

Code's  law  disproved 24.r 

Brorsen'i 


Clock,  its  error  and  rate 38 

Clusters  of  stars 304 

Colored  stars 298 

Colures,  equinoctial  and  solstitial 59 

Coma,  nucleus,  etc.,  of  comet 258 

Cometary  orbit  computed 265 

'     laws  of. 266 

Cometary  orbits,  nature  of 265 

"           "      position  of. 258 

Comet  defined 257 

"       known  to  be  periodic 268 

"       of  1744,  history 276 

"       of  1770,  history 276 

"       of  1770,  mass  of. 277 

"       of  1843,  history 278 

"       strike  the  earth 279 

Comet's  tail,  dimensions  of. 261 

"     formation  of 261 

"    position  of 261 

tails,  theory  of. 262 

Comets,  number  of. 257 

period  of  visibility 258 

phases  of. 264 

quantity  of  matter 264 

telescopic 263 

variations  of. 259 

with  several  tails 263 

Commensurability  of  periodic  times...  256 


comet 274|Cone  of  earth's  shadow 154 

Conic  sections,  heavenly  bodies  move 
in....  ...  138 


Calendar,  Gregorian 73 

"         Julian      72  Conjunction  and  opposition  defined...  11 

Cavendish's  experiment 29  "  "  of  a  planet  199 

Celestial  equator  defined 11  Constellations  enumerated 

"        globe,  problems  on 74 

"        sphere  defined 10 


names  of 

origin  of 


289 

287 
287 


342 


INDEX. 


Pape 

94 

175 

*% 

128 

27.", 
«» 
21 
28 
249 
281 
2S2 
2U3 
BO 
2U7 
86 
66 
207 
208 
78 
88 
80 
250 

251 
1!).-, 
I!).-, 
1110 
17 
12 
9 
11 
41 
1(1 
278 
291* 
297 

17 
14 
27 
27 
16 
61 
ir>,s 
18 
16 
68 
88 
64 
107 

65 
61 
157 
187 
108 
82 
1M 
1.-.4 

Pace 

155 
17 
173 
153 
327 
154 
171 
229 
172 
325 
169 
169 
172 
156 
100 
184 
183 
271 

55 
148 
108 
71 
28 
10 
59 
100 
185 
148 
148 

95 
34 
33 

274 
273 
284 
303 
284 
13.T 
117 

300 
199 
313 
36 
35 
128 
137 
67 
73 
103 

269 
270 
269 
11.) 
211 
311 
12 

"      surface,  irregularities  of.  
Eclipse  of  sun,  darkness  of  

Cotidal  lines            

Culminations  of  heavenly  bodies  

"        cvcle  of 

Day,  civil  and  astronomical...  

'        of  sun  and  moon  compared.. 

Density  of  earth  determined  
"       of  a  planet  determined  

'            "       different  kinds  

Differences  of  parallax  detected  

Ecliptic  limits  computed  

Electric  circuit  broken  by  clock  

Distance  between  two  stars  computed 
"        measured  by  sextant  
"        of  a  planet  from  sun  

"      hypothesis  considered  
Enlargement  of  sun  near  horizon  
Equation  of  the  centre  for  moon  
"           "         "       "    sun 

"       of  heavenly  body  computed 
"       of  the  sun  computed  
"        on  a  parallel  of  declination 

"           "     of  Jupiter  and  Sat- 

Eiurnal  inequality  in  North  Atlantic.  . 
"             "         in  Pacific 

"          precession  of.  

"             "         variations  of  

"        defined 

"           "        consequences  of  
"           "        described 

never  suspended  

Diurnal  paths  of  heavenly  bodies  

Earth,  circumference  of.  
"       dimensions  important  

"         "     classified  

Full  moon,  light  of  

"      ellipticitv  of 

"       proof  that  it  is  globular  
Earth's  annual  motion,  effects  of.  
"       atmosphere  effect  of.  

Geocentric  and  heliocentric  places.... 

Globes  artificial  

"       diameter,  how  determined.... 

Golden  number  

Gravitation  proportional  to  masses.... 

Gyroscope  illustrates  precession  

"     eccentricity  of.  

"       penumbra,  angle  of.  

"      peculiarities  of.  

'*              "        defined 

rotation,  direct  proof  of  

Heliocentric  longitude  of  a  planet  

w             "       formcf..... 

Horizon,  sensible  and  rational.... 

INDEX. 


343 


Horizontal  point  determined. 
Hour  circles  defined.... 


Inclination  of  a  planetary  orbit 

Inequalities  in  moon's  motion 

"           periodical  and  secular.... 
Inequality  of  solar  days 


Jupiter,  belts  of. 

' '       diameter 

"       distance  and  period 

"  spheroidal  form 

Jupiter's  belts  explained 

"  satellites,  configuration  of... 
"  distances  of 

"  "         mean  motions  of.. 


Kepler's  laws,  deductions  from. 

"        third  law 

"        three  laws 


Pmgei 

46JMars,  satellites  of 

20  "  spheroidal  form 

"  telescopic  appearance 

210  Mass  of  a  planet  determined 

150JMasses  of  sun  and  earth 

152|Measurements  of  meridian  arcs 

69  Mercury  and  Venus,  elongations 

"  "  "  phases  of. 

227  "  "  transits  of. 

227  "  greatest  brightness  of. 

227  period  and  distance 

227  "       rotation  on  axis 

228  Mercury,  visibility  of. 

231  Meridian  mark 

228  Meteoric  orbits 

232  Meteors  of  August  and  November 

Milky  way 

136  Minor  planets,  elements  of. 


]  35  Moon,  causes  no  refraction  ............. 


Latitude  andlongitude  defined 

"          "          "          of  a  star 

"        at  sea  determined ... 

"        of  a  place  determined 

"  "      "      how  known 

Law  of  gravitation  at  fixed  stars 

"  "  general 

Le Terrier  and  Adams's  researches... 

Librations  of  the  moon 

Light  of  sun  and  stars 

"      transmission  of. 

Line  of  the  apsides,  changes  of 

Local  times  compared 

Long  inequality  of  Jupiter 

"  "          ofUranus 

"     inequalities  of  planets 

Longitude  by  eclipses  of  moon 

"          "  electric  telegraph 

"  "  Jupiter's  satellites 

"  "  lunar  distances 

"  "  moon  culminating  stars 

"  "  solar  eclipse 

"          of  a  star  computed 

"          of  perihelion  of  a  planet . . . 

"          of  sun  computed 

Lunar  cycle 

"       day,  length  of. 

"       eclipses  computed 

"  "         diagram 

"  "         different  kinds 

"       ecliptic  limits 

"      mountains,  height  of. 

list  of. 

"      volcanoes  extinct 


129 

18 

60 
80 
80 
19 

r><>2 

137 

244 
124 
295 
232 

107 
183 
252 

2.V5 

*•>  ~  >> 

18( 
182 
231 
182 
181 
181 
86 

8 
12 


circular  craters 

diameter  of  ........................ 

distance  of.  ......................  — 

heat  of  .............................. 

revolution  of.  ...................... 

rotation  and  revolution  of.  ..... 

telescopic  appearance  ........... 

visible  in  a  solar  eclipse  ......... 

"      in  total  eclipse  ........... 

Moon's  atmosphere  ....................... 

centre  of  gravity  ................ 

diameter  affected  by  altitude 
disc,  obscure  part  ............... 

"     bright  points  on  .......... 

elliptic  motion  disturbed  ...... 

equatorial  parallax  ............. 

meridian  altitude  ............... 

nodes,  retrograde  motion  of.. 
orbit,  changes  of.  ............... 

"      eccentricity  of.  .......... 

"      form  of.  ................... 

path  about  the  sun  ............. 

"     position  of.  .......  .  ........ 

penumbra,  breadth  of.  ......... 

phases  ............................. 

rising,  daily  retardation  ....... 

rotation  upon  an  axis  .......... 

shadow,  breadth  of.  ............ 

"       length  of  .............. 

"       velocity  of.  ............ 

transits,  interval  of.  ............ 

Motion,  direct  and  retrograde  .......... 

of  the  solar  system  ............. 

on  a  curve,  law  of.  ............. 


330  Mural  circle  described 


Madler's  hypothesis 

Mars,  color  of 

"     distance  and  period. 

"     phases 


312 

223 

212' 


Nebula  crab 

"   dumb-bell .... 
in  Andromeds 

in  Lyra 

in  Orion 


223 
223 
223 
247 
143 

26 
213 
213 
217 
215 
214 
215 
215 

81 
280 
280 
310 
322 
116 
120 
100 
100 
118 
110 
125 
118 
174 
158 
116 
126 
115 
1H 
174 
143 
329 
113 
150 
127 
112 
112 
126 
112 
168 
113 
115 
124 
167 
166 
168 
113 
202 
206 
120 

43 

307 

307 
306 

308 
30C 


3-i-i 


INDEX. 


Nebulaspiral 

Nebulae,  brightness  of..., 

"       changes  of. 

"       described 

"       distribution  of., 


planetary 

' '  resolvable  into  stars 

Nebular  hypothesis,  argument  for..... 

"  "  stated 

Nebulosity  of  comets 

Nebulous  stars 

Neptune  discovered  at  Berlin 

"         distance  and  period 

' '         early  observations  of. 

"  satellite  of. 

Nodes  ascending  and  descending 

"  of  a  planetary  orbit 

"  of  the  moon's  orbit 

Nucleus,  dimensions  of 

Nutation,  solar  and  lunar 

Oblique  sphere  defined 

Obliquity  of  ecliptic  determined 

Observations  for  moon's  parallax 

"  made  in  meridian 

Occupations  of  Jupiter's  satellites 

"  of  stars 

Olbers's  hypothesis  inadmissible 

"  "  "  stated 

Orbit  of  a  binary  star 

Parallax  at  any  altitude  computed.... 
dependent  upon  ellipticity.. 

"        diurnal,  defined 

"         horizontal 

"         of  Alpha  Centauri 

"         of  fixed  stars,  catalogue 

"         of  moon  determined 

"        of  GlCygni 

of  various  stars 

' '         related  to  distance 

Parallel  sphere  defined 

Pendulum  experiments  in  a  mine 

"                "           on  a  mount- 
ain  

Periodical  comets,  elements  of. 

"         inequalities  of  planets 

Periodicity,  cause  of. 

Periodic  time  of  a  planet 

"        times  and  major  axes 

Perturbations  of  the  planets 

Planet  at  perihelion  fall  to  the  sun... 

"      deficient 

'      elements  of  orbit 

"      phases  of 

"      when  visible 

Planetary  system,  elements  of 

Planets,  apparent  motion  of 

diameter  of. 

*'       number  of...... 


Page! 

307  Planets,  orbits  of. 

310  Pole  of  equator,  motion  of. 

309     "    star,  how  found 

305|    "      "    varies 

309  Pons's  comet 

308 'Precession  of  equinoxes,  cause  of..... 

800          "          why  so  slow 

315jPressure  of  air  and  the  moon 

314|Problem  of  three  bodies 

2f>9 ,  Problems,  miscellaneous 

308 .  Progression  of  line  of  apsides 

244  Projectiles,  motion  of. 

245  Proper  motion  of  stars  explained 

244  Protuberances  emanate  from  sun 

245  flame-like 

110  "  nature  of. 

209 

127  Quadruple  stars 

258 

104  Reading  microscope  described 

Refraction  affected  by  temperature.... 

21          "          atmospheric 

64          "         determined  by  observation 

79          "         how  computed 

38  Resisting  medium 

230  Resultant  of  two  rotary  motions 

1 73  Revolution  near  earth's  surface 

226          "          of  celestial  sphere,  time  of 

224  Revolutions,  sidereal  and  synodic 

302  Right  ascension  defined. 

sphere  denned., 

Rotary  and  orbital  motions  of  earth.. 
79 
6  Satellites,  elements  of. 

76  "        motion  of.. 

293  "         number  of. 
33 1  Saturn,  diameter  of. 

77  "       distance  and  period  of. 

294  "       mass  and  density  of. 

294       "       rotation  of.... 

77  Saturn's  inner  ring, 

21        "       rings,  appearance  of. 

30  "  described 

dimensions  of. 

31  "  disappearance  of 

329        "  how  sustained 

252        "  observations  of 

291        "  variations  of. 

205        "       satellites. 

134  Secular  acceleration  of  moon 


constancy  of  major  axes 

inequalities  of  planets 

variation  of  eccentricity 

"       of  inclination 


201  Soxtant  described. 

204  Shooting  stars  described.... 


321        "           "     height  and  velocity  of 
200  Sidereal  and  solar  time  compared 


day  defined 

period,  how  determined 


199 
101 
13 
101 
275 
101 
103 
123 
142 
316 
151 
189 
296 
178 
177 
178 

304 

45 

55 

51 

53 

52 

271 

102 

140 

10 

110 

20 

21 

141 

324 

136 
199 
233 
233 
240 
233 
237 
238 
233 
234 
235 
238 
237 
234 
239 
152 

254: 

253 

254 

253 

49 

279 

279 

68 

11 

111 


INDEX. 


345 


Sidereal  time  defined  

(>7 
97 

17.-, 
170 

;»4 

96 

Tail  of  comet 

200 
99 
191 
190 
190 
189 
189 
186 
187 
186 
185 
187 
194 
196 
191 
192 
192 
193 
197 
198 
193 
194 
198 
190 
196 
193 
185 
188 
68 
81 
80 
83 
179 
47 
42 
39 
42 
40 
327 
218 
217 
330 
218 
218 
303 
60 
56 
59 
67 
57 
85 
122 

312 
241 
240 
241 
243 
241 
212 

Solar  atmosphere,  how  disturbed  
"     corona,  cause  of  
"     ecliptic  limits  computed  .    .    .  . 

Temperature  dependent  on  solar  spots 
Tidal  wave  a  forced  oscillation  

"     faculaj  are  elevations  

"        "     near  South  America  
"        "     origin  of.  

"     spots  and  terrestrial  storms  
'         "     black  nucleus  

'          '    changes  of  

!)() 
92 
9( 
91 
91 
92 

Tides  at  perigee  and  apogee 

'           '    depressions  

'          '     described  

'           '     latitude  of  

"     defined  

'     depend  on  moon's  declination.. 

.'{.•(I 
98 
246 

(17 

301 
300 

"         "    periodicity  of  

"     height  of.  

"     system  seen  from  Neptune  

'       near  projecting  promontory.... 
'       of  Gulf  of  Mexico  

"    70  Ophiuchi  

2sn 
289 
292 
311 
285 
2S<> 
288 
286 
299 
2!  to 
295 
291 
287 
14 
88 
.144 

92 
89 
98 
146 
144 
145 
92 
98 
89 
58 
58 
297 

of  rivers        .          

'     of  the  Mediterranean     

'     of  the  North  Atlantic  

"      distance  of  

'     of  the  Pacific  Ocean  

"     distribution  of.  

'     on  coast  of  Ireland  

"      diversity  of  brightness  of.  

'     why  complicated  

Time,  apparent  and  mean  

"      by  a  single  altitude  

4     optic<a  y    o 

"      of  sunrise  computed  

^     peno  ic.....  .^ 

"      under  different  meridians  
Transit  circle  described  

t     i     ^6 

'     twiiiklin     of 

"       instrument  adjusted  
"               "            described  
"               "            verified  

Sun's  attraction  a  disturbing  force.... 
"     disc  affected  by  refraction  

«'        of  Mercury,  catalogue  of.... 
"                "          intervals  of.  
"                 "            possible  

((              U       ft      ^            f 

"        of  Venus,  catalogue  of.  
"               "          intervals  of.  

"     disturbing  action  computed  
"             "             "      effect  of  
"             "         force  estimated  
"     envelope  not  solid  
"      equator,  position  of.  

1  ropus  an    po  a^ 

'  '     motion  in  declination  
"           "      in  right  ascension  

"             "         in  middle  latitudes 
"        end  of  computed  

"     parallax  and  distance  

221 
•22:} 
95 
55 
84 

Sunrise  affected  by  refraction  

"        discover}'  of.  
"        distance  and  period  of.  

Superior  planets  distinguished  
tynodic  period  determined  
"           "      of  a  planet  

111 
206 

^        irregulantie  ^ 

346 


INDEX. 


Variable  stars,  catalogue  of. 332 1  Water  on  the  moon 122 

Variation,  cause  of. 149  Weather  influenced  by  the  moon 123 

"  defined 1 49  Weight  at  pole  and  equator 25 

Velocity  of  a  body  in  a  curve 130  "  loss  of  at  equator 28 

Venus/morning  star 216 

period  and  distance  of. 2 1 6  Year,  anomalistic 109 

rotation  on  axis  of. . 

satellite  of 

twilight  of., 


21fi 
217 
21 C, 


beginning  of 74 

first  and  last  days  of 74 

tropical  and  sidereal 104 


visible  in  daytime 216 

Vernier  described 44  Zenith  and  nadir  defined 11 

"        of  graduated  circles 44  Zodiacal  light 99 

Vertical  circle  defined „...  ILZodiac,  signs  and  constellations  of....  101 

Volcanoes,  lunar 120        "         "     enumerated 60 

"          terrestrial 1 20  •  Zones  of  the  earth 61 

"         without  air  or  water 122J 


PLATE  T 


PLATE  II. 


PLATE  III. 


PLATE  VII. 


PLATE  VIII. 


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